Models Hydrodynamics 2nd class Results Proof
Anomalous fluctuations in one dimensional
interacting systems
Joint with
Júlia Komjáthy and Timo Seppäläinen
Márton Balázs
University of Bristol
Bath, 13 January, 2014.
Models Hydrodynamics 2nd class Results Proof
The models
Asymmetric simple exclusion process
Zero range
Bricklayers
Hydrodynamics
Characteristics
Tool: the second class particle
Single
Many second class particles
Results
Normal fluctuations
Abnormal fluctuations
Proof
Upper bound
Lower bound
Microscopic concavity/convexity
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦
◦
◦•
◦•
◦•
◦
◦•
◦
-3
-2
-1
0
1
2
3
4
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦
◦
◦•
◦•
◦•
◦
◦•
◦
-3
-2
-1
0
1
2
3
4
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦
◦
◦•
◦•
◦•
◦
•◦
◦
-3
-2
-1
0
1
2
3
4
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦
◦
◦•
◦•
◦• ◦
•◦
◦
-3
-2
-1
0
1
3
4
2
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦
◦
◦•
◦•
◦• ◦
◦•
◦
-3
-2
-1
0
1
3
4
2
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦
◦
◦•
◦•
◦• ◦
◦•
◦
-3
-2
-1
0
1
3
4
2
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦
◦
◦•
◦•
◦ • ◦
◦•
◦
-3
-2
-1
0
1
3
4
2
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦
◦
◦•
◦•
◦ •◦
◦•
◦
-3
-2
-1
0
1
3
4
2
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦
◦
◦•
◦•
◦ •◦
◦•
◦
-3
-2
-1
0
1
3
4
2
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦
◦
◦•
◦•
◦ •◦
•◦
◦
-3
-2
-1
0
1
3
4
2
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦
◦
◦•
◦•
◦
•◦
•◦
◦
-3
-2
-1
0
1
2
3
4
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦
◦
◦•
◦•
◦
◦•
◦•
◦
-3
-2
-1
0
1
2
3
4
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦
◦
◦•
◦•
◦
◦•
◦•
◦
-3
-2
-1
0
1
2
3
4
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦
◦
◦•
◦•
◦
◦•
◦• ◦
-3
-2
-1
0
1
2
3
4
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦
◦
◦•
◦•
◦
◦•
◦• ◦
-3
-2
-1
0
1
2
3
4
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦
◦
◦•
◦•
◦
◦•
◦• ◦
-3
-2
-1
0
1
2
3
4
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦
◦
◦•
◦•
◦
◦•
◦ • ◦
-3
-2
-1
0
1
2
3
4
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦
◦
◦•
◦•
◦
◦•
◦ •◦
-3
-2
-1
0
1
2
3
4
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦
◦
◦•
◦•
◦
◦•
◦ •◦
-3
-2
-1
0
1
2
3
4
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦
◦
◦•
◦•
◦
◦•
◦ •◦
-3
-2
-1
0
1
2
3
4
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦
◦
◦•
◦•
◦
◦•
◦
•◦
-3
-2
-1
0
1
2
3
4
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦
◦
◦•
◦•
◦
◦•
◦
•◦
-3
-2
-1
0
1
2
3
4
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦
◦
◦•
◦•
◦
◦•
◦
•◦
-3
-2
-1
0
1
2
3
4
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦
◦
◦•
◦•
◦
◦•
◦
•◦
-3
-2
-1
0
1
2
3
4
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦
◦
◦•
◦•
◦
◦•
◦
•◦
-3
-2
-1
0
1
2
3
4
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
• ◦
◦
◦•
◦•
◦
◦•
◦
•◦
-3
-2
-1
0
1
2
3
4
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
• ◦
◦
◦•
◦•
◦
◦•
◦
•◦
-3
-2
-1
0
1
2
3
4
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
•◦
◦
◦•
◦•
◦
◦•
◦
◦•
-3
-2
-1
0
1
2
3
4
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
•◦
◦
◦•
◦•
◦
◦•
◦
◦•
-3
-2
-1
0
1
2
3
4
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
•◦
◦
◦•
◦•
◦
◦•
◦
•◦
-3
-2
-1
0
1
2
3
4
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
•◦
◦
◦•
◦•
◦
◦•
◦
•◦
-3
-2
-1
0
1
2
3
4
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦•
◦
◦•
◦•
◦
◦•
◦
•◦
-3
-2
-1
0
1
2
3
4
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦•
◦
◦•
◦•
◦
◦•
◦
◦•
-3
-2
-1
0
1
2
3
4
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦•
◦
◦•
◦•
◦
◦•
◦
◦•
-3
-2
-1
0
1
2
3
4
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦•
◦
◦•
◦•
◦
◦•
◦
◦•
-3
-2
-1
0
1
2
3
4
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦•
◦
◦•
◦•
◦
◦•
◦
◦•
-3
-2
-1
0
1
2
3
4
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦•
◦
◦•
◦•
◦
◦•
◦
◦ •
-3
-2
-1
0
1
2
3
4
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦•
◦
◦•
◦•
◦
◦•
◦
◦ •
-3
-2
-1
0
1
2
3
4
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦•
◦
◦•
◦•
◦
◦•
◦
◦
-3
-2
-1
0
1
2
3
4
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦•
◦
◦•
◦•
◦
◦•
◦
◦
-3
-2
-1
0
1
2
3
4
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦•
◦
◦•
◦•
◦
◦•
◦
◦
-3
-2
-1
0
1
2
3
4
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦•
◦
◦•
◦•
◦
◦•
◦
◦
-3
-2
-1
0
1
2
3
4
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦•
◦
•◦
◦•
◦
•◦
◦
◦
-3
-2
-1
0
1
2
3
4
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦•
◦ •◦
◦•
◦
•◦
◦
◦
-3
-2
0
1
2
3
4
-1
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦•
◦ •◦
◦•
◦
◦•
◦
◦
-3
-2
0
1
2
3
4
-1
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦•
◦ •◦
◦•
◦
◦•
◦
◦
-3
-2
0
1
2
3
4
-1
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦•
◦ • ◦
◦•
◦
◦•
◦
◦
-3
-2
0
1
2
3
4
-1
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦•
◦• ◦
◦•
◦
◦•
◦
◦
-3
-2
0
1
2
3
4
-1
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦•
◦• ◦
◦•
◦
◦•
◦
◦
-3
-2
0
1
2
3
4
-1
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦•
◦• ◦
◦•
◦
•◦
◦
◦
-3
-2
0
1
2
3
4
-1
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦•
◦•
◦
◦•
◦
•◦
◦
◦
-3
-2
-1
0
1
2
3
4
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦•
◦•
◦
◦•
◦
◦•
◦
◦
-3
-2
-1
0
1
2
3
4
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Asymmetric simple exclusion
◦•
◦•
◦
◦•
◦
◦•
◦
◦
-3
-2
-1
0
1
2
3
4
i
Bernoulli(̺) distribution; ωi = 0 or 1.
Particles try to jump
to the right with rate p,
to the left with rate q = 1 − p < p.
The jump is suppressed if the destination site is occupied by
another particle.
The Bernoulli(̺) distribution is time-stationary for any
(0 ≤ ̺ ≤ 1). Any translation-invariant stationary distribution is a
mixture of Bernoullis.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric zero range process
••
-3
-2
•
•
•
•
-1
0
1
2
3
Poisson-type distribution; ωi ∈
4
Z+ .
i
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric zero range process
)
ω−3 =2
••
-3
-2
•
•
•
•
-1
0
1
2
3
Poisson-type distribution; ωi ∈
4
Z+ .
i
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric zero range process
)
ω−3 =2
••
-3
-2
•
•
•
•
-1
0
1
2
3
Poisson-type distribution; ωi ∈
Particles jump
to the right with rate p · r (ωi )
to the left with rate q · r (ωi )
4
i
Z+ .
(r non-decreasing)
(q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric zero range process
)
ω−3 =2
••
-3
-2
•
•
•
•
-1
0
1
2
3
Poisson-type distribution; ωi ∈
Particles jump
to the right with rate p · r (ωi )
to the left with rate q · r (ωi )
4
i
Z+ .
(r non-decreasing)
(q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric zero range process
)
ω−3 =2
•
•
• •
••
-3
-2
-1
0
1
2
3
Poisson-type distribution; ωi ∈
Particles jump
to the right with rate p · r (ωi )
to the left with rate q · r (ωi )
4
i
Z+ .
(r non-decreasing)
(q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric zero range process
)
ω−3 =2
•
•
• •
••
-3
-2
-1
0
1
2
3
Poisson-type distribution; ωi ∈
Particles jump
to the right with rate p · r (ωi )
to the left with rate q · r (ωi )
4
i
Z+ .
(r non-decreasing)
(q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric zero range process
)
ω−3 =2
•
•
• •
••
-3
-2
-1
0
1
2
3
Poisson-type distribution; ωi ∈
Particles jump
to the right with rate p · r (ωi )
to the left with rate q · r (ωi )
4
i
Z+ .
(r non-decreasing)
(q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric zero range process
)
ω−3 =2
••
-3
•
• • •
-2
-1
0
1
2
3
Poisson-type distribution; ωi ∈
Particles jump
to the right with rate p · r (ωi )
to the left with rate q · r (ωi )
4
i
Z+ .
(r non-decreasing)
(q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric zero range process
)
ω−3 =2
••
-3
•
• ••
-2
-1
0
1
2
3
Poisson-type distribution; ωi ∈
Particles jump
to the right with rate p · r (ωi )
to the left with rate q · r (ωi )
4
i
Z+ .
(r non-decreasing)
(q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric zero range process
)
ω−3 =2
••
-3
•
• ••
-2
-1
0
1
2
3
Poisson-type distribution; ωi ∈
Particles jump
to the right with rate p · r (ωi )
to the left with rate q · r (ωi )
4
i
Z+ .
(r non-decreasing)
(q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric zero range process
)
ω−3 =2
••
-3
•
• ••
-2
-1
0
1
2
3
Poisson-type distribution; ωi ∈
Particles jump
to the right with rate p · r (ωi )
to the left with rate q · r (ωi )
4
i
Z+ .
(r non-decreasing)
(q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric zero range process
)
ω−3 =2
••
-3
-2
•
•
••
-1
0
1
2
3
Poisson-type distribution; ωi ∈
Particles jump
to the right with rate p · r (ωi )
to the left with rate q · r (ωi )
4
i
Z+ .
(r non-decreasing)
(q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric zero range process
)
ω−3 =2
••
-3
-2
•
•
•
•
-1
0
1
2
3
Poisson-type distribution; ωi ∈
Particles jump
to the right with rate p · r (ωi )
to the left with rate q · r (ωi )
4
i
Z+ .
(r non-decreasing)
(q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric zero range process
)
ω−3 =2
••
-3
-2
••
•
•
-1
0
1
2
3
Poisson-type distribution; ωi ∈
Particles jump
to the right with rate p · r (ωi )
to the left with rate q · r (ωi )
4
i
Z+ .
(r non-decreasing)
(q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric zero range process
)
ω−3 =2
••
-3
-2
••
•
•
-1
0
1
2
3
Poisson-type distribution; ωi ∈
Particles jump
to the right with rate p · r (ωi )
to the left with rate q · r (ωi )
4
i
Z+ .
(r non-decreasing)
(q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric zero range process
)
ω−3 =2
••
-3
-2
••
•
•
-1
0
1
2
3
Poisson-type distribution; ωi ∈
Particles jump
to the right with rate p · r (ωi )
to the left with rate q · r (ωi )
4
i
Z+ .
(r non-decreasing)
(q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric zero range process
)
ω−3 =2
••
-3
-2
••
•
•
-1
0
1
2
3
Poisson-type distribution; ωi ∈
Particles jump
to the right with rate p · r (ωi )
to the left with rate q · r (ωi )
4
i
Z+ .
(r non-decreasing)
(q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric zero range process
)
ω−3 =2
••
-3
-2
• •
•
•
-1
0
1
2
3
Poisson-type distribution; ωi ∈
Particles jump
to the right with rate p · r (ωi )
to the left with rate q · r (ωi )
4
i
Z+ .
(r non-decreasing)
(q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric zero range process
)
ω−3 =2
••
-3
-2
• •
•
•
-1
0
1
2
3
Poisson-type distribution; ωi ∈
Particles jump
to the right with rate p · r (ωi )
to the left with rate q · r (ωi )
4
i
Z+ .
(r non-decreasing)
(q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric zero range process
)
ω−3 =2
••
-3
-2
• •
•
•
-1
0
1
2
3
Poisson-type distribution; ωi ∈
Particles jump
to the right with rate p · r (ωi )
to the left with rate q · r (ωi )
4
i
Z+ .
(r non-decreasing)
(q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric zero range process
)
••
-3
ω−3 =2
• •
-2
-1
•
•
0
1
2
3
Poisson-type distribution; ωi ∈
Particles jump
to the right with rate p · r (ωi )
to the left with rate q · r (ωi )
4
i
Z+ .
(r non-decreasing)
(q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric zero range process
)
••
-3
ω−3 =2
•
-2
•
••
-1
0
1
2
3
Poisson-type distribution; ωi ∈
Particles jump
to the right with rate p · r (ωi )
to the left with rate q · r (ωi )
4
i
Z+ .
(r non-decreasing)
(q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric zero range process
)
ω−3 =2
Particle current
=change in height
••
•
•
••
-3
-2
-1
0
h0
1
2
3
Poisson-type distribution; ωi ∈
Particles jump
to the right with rate p · r (ωi )
to the left with rate q · r (ωi )
4
i
Z+ .
(r non-decreasing)
(q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric zero range process
)
ω−3 =2
Particle current
=change in height
••
•
•
••
-3
-2
-1
0
h0
1
2
3
Poisson-type distribution; ωi ∈
Particles jump
to the right with rate p · r (ωi )
to the left with rate q · r (ωi )
4
i
Z+ .
(r non-decreasing)
(q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric zero range process
)
ω−3 =2
Particle current
=change in height
••
•
•
••
-3
-2
-1
0
h0
1
2
3
Poisson-type distribution; ωi ∈
Particles jump
to the right with rate p · r (ωi )
to the left with rate q · r (ωi )
4
i
Z+ .
(r non-decreasing)
(q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric zero range process
)
ω−3 =2
Particle current
=change in height
••
•
•
••
-3
-2
-1
0
h0
1
2
3
Poisson-type distribution; ωi ∈
Particles jump
to the right with rate p · r (ωi )
to the left with rate q · r (ωi )
4
i
Z+ .
(r non-decreasing)
(q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric zero range process
)
ω−3 =2
Particle current
=change in height
••
•
•
••
-3
-2
-1
0
h0
1
2
3
Poisson-type distribution; ωi ∈
Particles jump
to the right with rate p · r (ωi )
to the left with rate q · r (ωi )
4
i
Z+ .
(r non-decreasing)
(q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric zero range process
)
ω−3 =2
Particle current
=change in height
h0
••
•
•
• •
-3
-2
-1
0
1
2
3
Poisson-type distribution; ωi ∈
Particles jump
to the right with rate p · r (ωi )
to the left with rate q · r (ωi )
4
i
Z+ .
(r non-decreasing)
(q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric zero range process
)
ω−3 =2
Particle current
=change in height
h0
••
•
•
• •
-3
-2
-1
0
1
2
3
Poisson-type distribution; ωi ∈
Particles jump
to the right with rate p · r (ωi )
to the left with rate q · r (ωi )
4
i
Z+ .
(r non-decreasing)
(q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric zero range process
)
ω−3 =2
Particle current
=change in height
h0
••
•
•
• •
-3
-2
-1
0
1
2
3
Poisson-type distribution; ωi ∈
Particles jump
to the right with rate p · r (ωi )
to the left with rate q · r (ωi )
4
i
Z+ .
(r non-decreasing)
(q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric zero range process
)
ω−3 =2
Particle current
=change in height
h0
••
•
•
• •
-3
-2
-1
0
1
2
3
Poisson-type distribution; ωi ∈
Particles jump
to the right with rate p · r (ωi )
to the left with rate q · r (ωi )
4
i
Z+ .
(r non-decreasing)
(q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric zero range process
)
ω−3 =2
Particle current
=change in height
••
•
•
•
-3
-2
-1
0
h0
•
1
2
3
Poisson-type distribution; ωi ∈
Particles jump
to the right with rate p · r (ωi )
to the left with rate q · r (ωi )
4
i
Z+ .
(r non-decreasing)
(q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric zero range process
)
ω−3 =2
Particle current
=change in height
••
•
•
•
-3
-2
-1
0
h0
•
1
2
3
Poisson-type distribution; ωi ∈
Particles jump
to the right with rate p · r (ωi )
to the left with rate q · r (ωi )
4
i
Z+ .
(r non-decreasing)
(q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
ωi+1 =−1
ωi
ωi+1
|| 1
−1
i +1
Poisson-type distribution; ωi ∈ Z.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
ωi+1 =−1
ωi
ωi+1
|| 1
−1
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
ωi+1 =−1
ωi
ωi+1
|| 1
−1
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r(ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r(−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
ωi+1 =−1
ωi
ωi+1
|| 1
−1
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r(ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r(−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
ωi+1 =−1
ωi
ωi+1
|| 1
−1
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r(ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r(−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
ωi+1 =−1
ωi
ωi+1
|| 1
−1
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r(ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r(−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
ωi+1 =−1
ωi
ωi+1
|| 1
−1
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r(ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r(−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
ωi+1 =−1
ωi
ωi+1
|| 1
−1
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r(ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r(−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
ωi+1 =−1
ωi
ωi+1
|| 1
−1
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r(ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r(−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
ωi+1 =−1
ωi
ωi+1
|| 1
−1
i+1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r(−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r(ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
ωi+1 =−1
ωi
ωi+1
|| 1
−1
i+1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r(−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r(ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
ωi+1 =−1
ωi
ωi+1
|| 1
−1
i+1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r(−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r(ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
ωi+1 =−1
ωi
ωi+1
|| 1
−1
i+1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r(−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r(ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
ωi+1 =−1
ωi
ωi+1
|| 1
−1
i+1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r(−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r(ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
ωi+1 =−1
ωi
ωi+1
|| 1
−1
i+1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r(−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r(ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
ωi+1 =−1
ωi
ωi+1
|| 1
−1
i+1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r(−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r(ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
ωi
ωi − 1
→
ωi+1 + 1
ωi+1
|| ||
1
0
→
−1
0
i
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
ωi
ωi − 1
→
ωi+1 + 1
ωi+1
|| ||
1
0
→
−1
0
i
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
ωi
ωi − 1
→
ωi+1 + 1
ωi+1
|| ||
1
0
→
−1
0
i
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
ωi
ωi − 1
→
ωi+1 + 1
ωi+1
|| ||
1
0
→
−1
0
i
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
ωi
ωi − 1
→
ωi+1 + 1
ωi+1
|| ||
1
0
→
−1
0
i
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
ωi
ωi − 1
→
ωi+1 + 1
ωi+1
|| ||
1
0
→
−1
0
i
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
ωi
ωi − 1
→
ωi+1 + 1
ωi+1
|| ||
1
0
→
−1
0
i
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
ωi
ωi − 1
→
ωi+1 + 1
ωi+1
|| ||
1
0
→
−1
0
i
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
ωi
ωi − 1
→
ωi+1 + 1
ωi+1
|| ||
1
0
→
−1
0
i
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
ωi
ωi − 1
→
ωi+1 + 1
ωi+1
|| ||
1
0
→
−1
0
i
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The asymmetric bricklayers process
i
i +1
Poisson-type distribution; ωi ∈ Z.
a brick is added with rate p · [r (ωi ) + r (−ωi+1 )]
a brick is removed with rate q · [r (−ωi ) + r (ωi+1 )].
(r (ω) · r (1 − ω) = 1;
r non-decreasing; q = 1 − p < p).
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The model
ωi − 1
ωi+1 + 1
ωi
ωi + 1
→
ωi+1 − 1
ωi+1
◮
ωi
ωi+1
→
with rate p(ωi , ωi+1 ),
with rate q(ωi , ωi+1 ), where
p and q are such that they keep the state space (ASEP,
ZRP),
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The model
ωi − 1
ωi+1 + 1
ωi
ωi + 1
→
ωi+1 − 1
ωi+1
◮
ωi
ωi+1
→
with rate p(ωi , ωi+1 ),
with rate q(ωi , ωi+1 ), where
p and q are such that they keep the state space (ASEP,
ZRP),
◮
p is non-decreasing in the first, non-increasing in the
second variable, and q vice-versa (attractivity),
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The model
ωi − 1
ωi+1 + 1
ωi
ωi + 1
→
ωi+1 − 1
ωi+1
◮
ωi
ωi+1
→
with rate p(ωi , ωi+1 ),
with rate q(ωi , ωi+1 ), where
p and q are such that they keep the state space (ASEP,
ZRP),
◮
p is non-decreasing in the first, non-increasing in the
second variable, and q vice-versa (attractivity),
◮
they satisfy some algebraic conditions to get a product
stationary distribution for the process,
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The model
ωi − 1
ωi+1 + 1
ωi
ωi + 1
→
ωi+1 − 1
ωi+1
◮
ωi
ωi+1
→
with rate p(ωi , ωi+1 ),
with rate q(ωi , ωi+1 ), where
p and q are such that they keep the state space (ASEP,
ZRP),
◮
p is non-decreasing in the first, non-increasing in the
second variable, and q vice-versa (attractivity),
◮
they satisfy some algebraic conditions to get a product
stationary distribution for the process,
◮
they satisfy some regularity conditions to make sure the
dynamics exists.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Integrated particle current
h
t=0
i
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Integrated particle current
h
0
0
t=0
i
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Integrated particle current
h t
t
0
0
t=0
i
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Integrated particle current
h t
t
Vt
0
0
t=0
i
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
Integrated particle current
h t
hVt (t)
t
Vt
0
0
t=0
i
hVt (t) = height as seen by a moving observer of velocity V .
= net number of particles passing the window s 7→ Vs.
(Remember: particle current=change in height.)
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The question
... is the properties of hVt (t) under the time-stationary evolution.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The question
... is the properties of hVt (t) under the time-stationary evolution.
◮
E(hVt (t)) = t · E(growth rate) is easily computed with
martingales.
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The question
... is the properties of hVt (t) under the time-stationary evolution.
◮
E(hVt (t)) = t · E(growth rate) is easily computed with
martingales.
◮
Law of Large Numbers:
ergodicity arguments.
hVt (t)
−→
t
t→∞
E(growth rate) by
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The question
... is the properties of hVt (t) under the time-stationary evolution.
◮
E(hVt (t)) = t · E(growth rate) is easily computed with
martingales.
◮
Law of Large Numbers:
hVt (t)
−→
t
t→∞
E(growth rate) by
ergodicity arguments.
◮
Var(hVt (t))? That is, time-order and scaling limit? Central
Limit Theorem, if relevant at all?
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The question
... is the properties of hVt (t) under the time-stationary evolution.
◮
E(hVt (t)) = t · E(growth rate) is easily computed with
martingales.
◮
Law of Large Numbers:
hVt (t)
−→
t
t→∞
E(growth rate) by
ergodicity arguments.
◮
Var(hVt (t))? That is, time-order and scaling limit? Central
Limit Theorem, if relevant at all?
◮
Distributional limit of hVt (t) in the correct scaling?
Models Hydrodynamics 2nd class Results Proof
ASEP AZRP ABLP
The question
... is the properties of hVt (t) under the time-stationary evolution.
◮
E(hVt (t)) = t · E(growth rate) is easily computed with
martingales.
◮
Law of Large Numbers:
hVt (t)
−→
t
t→∞
E(growth rate) by
ergodicity arguments.
◮
Var(hVt (t))? That is, time-order and scaling limit? Central
Limit Theorem, if relevant at all?
◮
Distributional limit of hVt (t) in the correct scaling?
Models Hydrodynamics 2nd class Results Proof
Characteristics
Hydrodynamics (very briefly)
The density ̺ : = E(ω) and the hydrodynamic flux
H : = E[growth rate] both depend on a parameter of the
stationary distribution.
Models Hydrodynamics 2nd class Results Proof
Characteristics
Hydrodynamics (very briefly)
The density ̺ : = E(ω) and the hydrodynamic flux
H : = E[growth rate] both depend on a parameter of the
stationary distribution.
◮
H(̺) is the hydrodynamic flux function.
Models Hydrodynamics 2nd class Results Proof
Characteristics
Hydrodynamics (very briefly)
The density ̺ : = E(ω) and the hydrodynamic flux
H : = E[growth rate] both depend on a parameter of the
stationary distribution.
◮
H(̺) is the hydrodynamic flux function.
◮
If the process is locally in equilibrium, but changes over
some large scale (variables X = εi and T = εt), then
∂T ̺(T , X ) + ∂X H(̺(T , X )) = 0
(conservation law).
Models Hydrodynamics 2nd class Results Proof
Characteristics
Hydrodynamics (very briefly)
The density ̺ : = E(ω) and the hydrodynamic flux
H : = E[growth rate] both depend on a parameter of the
stationary distribution.
◮
H(̺) is the hydrodynamic flux function.
◮
If the process is locally in equilibrium, but changes over
some large scale (variables X = εi and T = εt), then
∂T ̺(T , X ) + ∂X H(̺(T , X )) = 0
◮
(conservation law).
The characteristics is a path X (T ) where ̺(T , X (T )) is
constant.
Models Hydrodynamics 2nd class Results Proof
Characteristics
Characteristics (very briefly)
∂T ̺ + ∂X H(̺) = 0
Models Hydrodynamics 2nd class Results Proof
Characteristics
Characteristics (very briefly)
∂T ̺ + ∂X H(̺) = 0
∂T ̺ + H ′ (̺) · ∂X ̺ = 0
(while smooth)
Models Hydrodynamics 2nd class Results Proof
Characteristics
Characteristics (very briefly)
∂T ̺ + ∂X H(̺) = 0
∂T ̺ + H ′ (̺) · ∂X ̺ = 0
(while smooth)
d
̺(T , X (T )) = 0
dT
Models Hydrodynamics 2nd class Results Proof
Characteristics
Characteristics (very briefly)
∂T ̺ + ∂X H(̺) = 0
∂T ̺ + H ′ (̺) · ∂X ̺ = 0
∂T ̺ + Ẋ (T ) · ∂X ̺ =
(while smooth)
d
̺(T , X (T )) = 0
dT
Models Hydrodynamics 2nd class Results Proof
Characteristics
Characteristics (very briefly)
∂T ̺ + ∂X H(̺) = 0
∂T ̺ + H ′ (̺) · ∂X ̺ = 0
∂T ̺ + Ẋ (T ) · ∂X ̺ =
(while smooth)
d
̺(T , X (T )) = 0
dT
Models Hydrodynamics 2nd class Results Proof
Characteristics
Characteristics (very briefly)
∂T ̺ + ∂X H(̺) = 0
∂T ̺ + H ′ (̺) · ∂X ̺ = 0
∂T ̺ + Ẋ (T ) · ∂X ̺ =
(while smooth)
d
̺(T , X (T )) = 0
dT
So, Ẋ (T ) = H ′ (̺) = : C is the characteristic speed.
Models Hydrodynamics 2nd class Results Proof
Characteristics
Characteristics (very briefly)
∂T ̺ + ∂X H(̺) = 0
∂T ̺ + H ′ (̺) · ∂X ̺ = 0
∂T ̺ + Ẋ (T ) · ∂X ̺ =
(while smooth)
d
̺(T , X (T )) = 0
dT
So, Ẋ (T ) = H ′ (̺) = : C is the characteristic speed.
If H(̺) is convex or concave, then the Rankine-Hugoniot speed
for densities ̺ and λ is
R=
H(̺) − H(λ)
.
̺−λ
Models Hydrodynamics 2nd class Results Proof
Characteristics
Characteristics (very briefly)
∂T ̺ + ∂X H(̺) = 0
∂T ̺ + H ′ (̺) · ∂X ̺ = 0
∂T ̺ + Ẋ (T ) · ∂X ̺ =
(while smooth)
d
̺(T , X (T )) = 0
dT
So, Ẋ (T ) = H ′ (̺) = : C is the characteristic speed.
If H(̺) is convex or concave, then the Rankine-Hugoniot speed
for densities ̺ and λ is
R=
H(̺) − H(λ)
.
̺−λ
This would be the speed of a shock of densities ̺ and λ.
Models Hydrodynamics 2nd class Results Proof
Characteristics
Characteristics (very briefly)
∂T ̺ + ∂X H(̺) = 0
∂T ̺ + H ′ (̺) · ∂X ̺ = 0
∂T ̺ + Ẋ (T ) · ∂X ̺ =
(while smooth)
d
̺(T , X (T )) = 0
dT
So, Ẋ (T ) = H ′ (̺) = : C is the characteristic speed.
If H(̺) is convex or concave, then the Rankine-Hugoniot speed
for densities ̺ and λ is
R=
H(̺) − H(λ)
.
̺−λ
This would be the speed of a shock of densities ̺ and λ.
C = H ′ (̺)
R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Characteristics
Characteristics (very briefly)
Convex flux (some cases of AZRP, ABLP):
H(̺)
̺
C = H ′ (̺)
R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Characteristics
Characteristics (very briefly)
Convex flux (some cases of AZRP, ABLP):
H(̺)
̺
C = H ′ (̺)
̺
R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Characteristics
Characteristics (very briefly)
Convex flux (some cases of AZRP, ABLP):
H(̺)
C
̺
̺
C = H ′ (̺)
C = H ′ (̺)
R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Characteristics
Characteristics (very briefly)
Convex flux (some cases of AZRP, ABLP):
H(̺)
R
C
̺
λ
C = H ′ (̺)
C = H ′ (̺)
R=
̺
H(̺) − H(λ)
̺−λ
R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Characteristics
Characteristics (very briefly)
Convex flux (some cases of AZRP, ABLP):
H(̺)
R
C
λ
C = H ′ (̺) > R =
C = H ′ (̺)
̺
̺
H(̺) − H(λ)
̺−λ
R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Characteristics
Characteristics (very briefly)
Concave flux (ASEP, AZRP):
H(̺)
̺
C = H ′ (̺)
R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Characteristics
Characteristics (very briefly)
Concave flux (ASEP, AZRP):
H(̺)
̺
C = H ′ (̺)
̺
R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Characteristics
Characteristics (very briefly)
Concave flux (ASEP, AZRP):
C
H(̺)
̺
̺
C = H ′ (̺)
C = H ′ (̺)
R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Characteristics
Characteristics (very briefly)
Concave flux (ASEP, AZRP):
C
H(̺)
R
̺
λ
C = H ′ (̺)
C = H ′ (̺)
R=
̺
H(̺) − H(λ)
̺−λ
R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Characteristics
Characteristics (very briefly)
Concave flux (ASEP, AZRP):
C
H(̺)
R
λ
C = H ′ (̺) < R =
C = H ′ (̺)
<
̺
̺
H(̺) − H(λ)
̺−λ
R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
•
•
•
•
•
•
•
•
•
••
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
•
•
•
•
•
•
•
•
•
••
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the right:
rate≤rate
•
•
•
•
•
•
•
•
•
••
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the right:
rate≤rate
with rate:
•
•
•
•
•
•
•
•
•
••
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the right:
rate≤rate
with rate:
•
•
•
•
••
•
•
•
•
•
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the right:
rate≤rate
with rate:
•
•
•
•
••
•
•
•
•
•
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the right:
rate≤rate
with rate:
•
•
•
•
•
•• •
•
• •
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the right:
rate≤rate
with rate:
•
•
•
•
•
•• •
•
• •
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the right:
rate≤rate
with rate:
•
•
•
•
• •
•
•
•
•
•
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the right:
rate≤rate
with rate:
•
•
•
•
•
• ••
•
••
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the right:
rate≤rate
with rate:
•
•
•
•
•
• ••
•
••
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the right:
rate≤rate
with rate:
•
•
•
•
•
• ••
•
••
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the right:
rate≤rate
with rate:
•
•
•
•
•
•
••
•
••
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the right:
rate≤rate
with rate:
•
•
•
•
•
•
•
•
•
•
•
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the right:
rate≤rate
with rate-rate:
•
•
•
•
•
•
•
•
•
••
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the right:
rate≤rate
with rate-rate:
•
•
•
•
••
•
•
•
••
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the right:
rate≤rate
with rate-rate:
•
•
•
•
••
•
•
•
••
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the right:
rate≤rate
with rate-rate:
•
•
•
•
•
•• •
• ••
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the right:
rate≤rate
with rate-rate:
•
•
•
•
•
•• •
• ••
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the right:
rate≤rate
with rate-rate:
•
•
•
•
•
• • •
• ••
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the right:
rate≤rate
with rate-rate:
•
•
•
•
•
• ••
• ••
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the right:
rate≤rate
with rate-rate:
•
•
•
•
•
• ••
• ••
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the right:
rate≤rate
with rate-rate:
•
•
•
•
•
• ••
• ••
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the right:
rate≤rate
with rate-rate:
•
•
•
•
•
•
•
••
••
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the right:
rate≤rate
with rate-rate:
•
•
•
•
•
•
•
•
•
••
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the left:
rate≥rate
•
•
•
•
•
•
•
•
•
••
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the left:
rate≥rate
with rate:
•
•
•
•
•
•
•
•
•
••
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the left:
rate≥rate
with rate:
••
••
•
•
•
••
••
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the left:
rate≥rate
with rate:
•
•• •
•• •
••
••
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the left:
rate≥rate
with rate:
•
•• •
•• •
••
••
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the left:
rate≥rate
with rate:
•
•• •
•• •
••
••
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the left:
rate≥rate
with rate:
•
• • •
• • •
••
••
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the left:
rate≥rate
with rate:
•
• ••
• ••
••
••
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the left:
rate≥rate
with rate:
•
• ••
• ••
••
••
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the left:
rate≥rate
with rate:
•
• ••
• ••
••
••
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the left:
rate≥rate
with rate:
•
•
•
••
••
••
••
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the left:
rate≥rate
with rate:
•
•
•
•
•
•
•
••
••
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the left:
rate≥rate
with rate-rate:
•
•
•
•
•
•
•
•
•
••
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the left:
rate≥rate
with rate-rate:
•
•
••
•
•
•
•
•
•
•
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the left:
rate≥rate
with rate-rate:
•
• •
•
•
•
•
•
•
•
•
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the left:
rate≥rate
with rate-rate:
•
• •
•
•
•
•
•
•
•
•
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the left:
rate≥rate
with rate-rate:
•
• •
•
•
•
•
•
•
•
•
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the left:
rate≥rate
with rate-rate:
•
• •
•
•
•
•
•
•
•
•
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the left:
rate≥rate
with rate-rate:
•
• •
•
•
•
•
•
•
•
•
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the left:
rate≥rate
with rate-rate:
•
• •
•
•
•
•
•
•
•
•
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the left:
rate≥rate
with rate-rate:
•
• •
•
•
•
•
•
•
•
•
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the left:
rate≥rate
with rate-rate:
•
•
•
•
•
••
•
•
•
•
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the left:
rate≥rate
with rate-rate:
•
•
•
•
•
•
•
•
•
•
•
i
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
States ω and ω only differ at one site.
Growth on the left:
rate≥rate
with rate-rate:
•
•
•
•
•
•
•
•
•
•
•
i
A single discrepancy , the second class particle, is conserved.
Its position at time t is Q(t).
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
Theorem (B. - Seppäläinen; also ideas from B. Tóth, H. Spohn
and M. Prähofer)
Started from (almost) equilibrium,
E(Q(t)) = C · t
in the whole family of processes.
C = H ′ (̺)
<
R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
Theorem (B. - Seppäläinen; also ideas from B. Tóth, H. Spohn
and M. Prähofer)
Started from (almost) equilibrium,
E(Q(t)) = C · t
in the whole family of processes.
C is the characteristic speed.
C = H ′ (̺)
<
R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
Theorem (B. - Seppäläinen; also ideas from B. Tóth, H. Spohn
and M. Prähofer)
Started from (almost) equilibrium,
E(Q(t)) = C · t
in the whole family of processes.
C is the characteristic speed.
The second class particle follows the characteristics, people
have known this for a long time.
C = H ′ (̺)
<
R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Tool: the second class particle
Theorem (B. - Seppäläinen; also ideas from B. Tóth, H. Spohn
and M. Prähofer)
Started from (almost) equilibrium,
E(Q(t)) = C · t
in the whole family of processes.
C is the characteristic speed.
The second class particle follows the characteristics, people
have known this for a long time.
C = H ′ (̺) = EQ/t
<
R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles
λ
•
•
-5
C = H ′ (̺) = EQ/t
•
•
•
5
0
<
•
•
10
15
i
R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second
class particles
̺
λ
-1 0
-2
•
•
•
•
•
•
•
-5
C = H ′ (̺) = EQ/t
1
2
•
•
•
•
•
•
•
•
•
10
15
5
0
<
•
•
•
i
R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second
class particles
̺
λ
-1 0
-2
•
•
•
•
•
•
•
-5
C = H ′ (̺) = EQ/t
1
2
•
•
•
•
•
•
•
•
•
10
15
5
0
<
•
•
•
i
R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second
class particles
̺
λ
-1 0
-2
•
•
•
•
•
•
•
-5
C = H ′ (̺) = EQ/t
1
2
•
•
•
•
•
•
•
•
•
10
15
5
0
<
•
•
•
i
R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second
class particles
̺
λ
-1 0
-2
•
•
•
•
•
•
•
-5
C = H ′ (̺) = EQ/t
1
2
•
•
•
•
•
•
•
•
•
10
15
5
0
<
•
•
•
i
R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second
class particles
̺
λ
-1 0
-2
•
•
•
•
•
•
•
-5
C = H ′ (̺) = EQ/t
1
2
•
••
•
•
••
•
•
10
15
5
0
<
•
•
•
i
R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second
class particles
̺
λ
-1 0
-2
•
•
•
•
•
•
•
-5
C = H ′ (̺) = EQ/t
1
2
•
••
•
•
••
•
•
10
15
5
0
<
•
•
•
i
R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second
class particles
̺
λ
-1 0
-2
•
•
•
•
•
•
•
-5
C = H ′ (̺) = EQ/t
1
2
•
••
•
•
••
•
•
10
15
5
0
<
•
•
•
i
R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second
class particles
̺
λ
-1 0
-2
•
•
•
•
•
•
•
-5
C = H ′ (̺) = EQ/t
1
2
•
• •
•
•
• •
•
•
10
15
5
0
<
•
•
•
i
R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second
class particles
̺
λ
-1 0
-2
•
•
•
•
•
•
•
-5
C = H ′ (̺) = EQ/t
1
2
•
• •
•
•
• •
•
•
10
15
5
0
<
•
•
•
i
R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second
class particles
̺
λ
-1 0
-2
•
•
•
•
•
•
•
-5
C = H ′ (̺) = EQ/t
1
2
•
• •
•
•
• •
•
•
10
15
5
0
<
•
•
•
i
R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second
class particles
̺
λ
-1 0
-2
•
•
•
•
•
•
•
-5
C = H ′ (̺) = EQ/t
1
•
•
•
•
•
•
•
•
•
10
15
5
0
<
•
2
•
•
i
R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second
class particles
̺
λ
-1 0
-2
•
•
•
•
•
•
•
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R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
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R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
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R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
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R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
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R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
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R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
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R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
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R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
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15
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R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
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R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
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R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
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R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
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R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
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R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
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15
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R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
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15
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R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
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R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
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R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
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R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
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R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
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Models Hydrodynamics 2nd class Results Proof
Single Many
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R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
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Models Hydrodynamics 2nd class Results Proof
Single Many
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Models Hydrodynamics 2nd class Results Proof
Single Many
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Models Hydrodynamics 2nd class Results Proof
Single Many
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Models Hydrodynamics 2nd class Results Proof
Single Many
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Models Hydrodynamics 2nd class Results Proof
Single Many
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Models Hydrodynamics 2nd class Results Proof
Single Many
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Models Hydrodynamics 2nd class Results Proof
Single Many
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Models Hydrodynamics 2nd class Results Proof
Single Many
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Models Hydrodynamics 2nd class Results Proof
Single Many
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Models Hydrodynamics 2nd class Results Proof
Single Many
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R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
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R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
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Models Hydrodynamics 2nd class Results Proof
Single Many
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R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
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Models Hydrodynamics 2nd class Results Proof
Single Many
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R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
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R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
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R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
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15
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R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
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15
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R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
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15
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R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
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15
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R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
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15
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R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
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15
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R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
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15
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R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
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15
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R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
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15
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R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
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15
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R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
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0
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10
15
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R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
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0
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•
10
15
i
R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second
class particles
̺
λ
0
-1
-2
•
•
•
••
•
•
•
•
-5
C = H ′ (̺) = EQ/t
1
5
0
<
•
2
•
•
•
•
•
•
•
•
•
10
15
i
R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second
class particles
̺
λ
0
-1
-2
•
•
•
••
•
•
•
•
-5
C = H ′ (̺) = EQ/t
1
5
0
<
•
2
•
•
•
•
•
•
•
•
•
10
15
i
R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second
class particles
̺
λ
0
-1
-2
•
•
•
••
•
•
•
•
-5
C = H ′ (̺) = EQ/t
1
5
0
<
•
2
•
•
•
•
•
•
•
•
•
10
15
i
R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second
class particles
̺
λ
-2
•
-1
•
•
••
•
•
•
•
-5
C = H ′ (̺) = EQ/t
0
1
5
0
<
•
2
•
•
•
•
•
•
•
•
•
10
15
i
R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second
class particles
̺
λ
-2
•
-1
•
•
• •
•
•
•
•
-5
C = H ′ (̺) = EQ/t
0
1
5
0
<
•
2
•
•
•
•
•
•
•
•
•
10
15
i
R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second
class particles
̺
λ
-1 0
-2
•
•
•
• •
•
•
•
•
-5
C = H ′ (̺) = EQ/t
1
5
0
<
•
2
•
•
•
•
•
•
•
•
•
10
15
i
R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second
class particles
̺
λ
-1 0
-2
•
•
•
•
• •
•
•
•
-5
C = H ′ (̺) = EQ/t
1
5
0
<
•
2
•
•
•
•
•
•
•
•
•
10
15
i
R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second
class particles
̺
λ
-1 0
-2
•
•
•
•
•
•
•
•
-5
C = H ′ (̺) = EQ/t
1
•
5
0
<
•
2
•
•
•
•
•
•
•
•
•
10
15
i
R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second
class particles
̺
λ
-1 0
-2
•
•
•
•
•
•
•
•
-5
Picture:
The position X (t) of
′
C = H (̺) = EQ/t
•
5
0
0
1
•
2
•
•
•
•
•
•
•
•
•
10
15
i
follows the Rankine-Hugoniot speed R.
<
R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second
class particles
̺
λ
-1 0
-2
•
•
•
•
•
•
•
•
-5
Picture:
The position X (t) of
′
C = H (̺) = EQ/t
•
5
0
0
1
•
2
•
•
•
•
•
•
•
•
•
10
15
i
follows the Rankine-Hugoniot speed R.
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Characteristics (very briefly)
Convex flux (some cases of AZRP, ABLP):
H(̺)
R
C
̺
λ
Recall
C = H ′ (̺) = EQ/t
C = H ′ (̺) > R =
̺
H(̺) − H(λ)
̺−λ
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Characteristics (very briefly)
Convex flux (some cases of AZRP, ABLP):
H(̺)
R
C
̺
λ
Recall
C = H ′ (̺) > R =
Do we have
C = H ′ (̺) = EQ/t
̺
H(̺) − H(λ)
̺−λ
?
Q(t) ≥ X (t)
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Characteristics (very briefly)
Convex flux (some cases of AZRP, ABLP):
H(̺)
R
C
̺
λ
Recall
C = H ′ (̺) > R =
Do we have
C = H ′ (̺) = EQ/t
̺
H(̺) − H(λ)
̺−λ
?
Q(t) ≥ X (t) − tight error
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Characteristics (very briefly)
Concave flux (ASEP, AZRP):
C
H(̺)
R
̺
λ
C = H ′ (̺) < R =
C = H ′ (̺) = EQ/t
<
̺
H(̺) − H(λ)
̺−λ
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Characteristics (very briefly)
Concave flux (ASEP, AZRP):
C
H(̺)
R
̺
λ
C = H ′ (̺) < R =
H(̺) − H(λ)
̺−λ
?
Q(t) ≤ X (t)
Do we have
C = H ′ (̺) = EQ/t
̺
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Characteristics (very briefly)
Concave flux (ASEP, AZRP):
C
H(̺)
R
̺
λ
C = H ′ (̺) < R =
H(̺) − H(λ)
̺−λ
?
Q(t) ≤ X (t) + tight error
Do we have
C = H ′ (̺) = EQ/t
̺
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles
λ
•
•
-5
C = H ′ (̺) = EQ/t
•
•
•
5
0
<
•
•
10
15
i
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles
̺
λ
-1 0
-2
•
•
•
•
•
•
•
-5
C = H ′ (̺) = EQ/t
1
2
•
•
•
•
•
•
•
•
•
10
15
5
0
<
•
•
•
i
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-1 0
-2
•
•
•
•
•
•
•
•
-5
•
•
•
1
•
•
•
•
•
•
•
•
•
5
0
2
•
•
•
•
•
•
•
•
•
•
10
15
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-1 0
-2
•
•
•
•
•
•
•
•
-5
•
•
•
1
•
•
•
•
•
•
•
•
•
5
0
2
•
•
•
•
•
•
•
•
•
•
10
15
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-1 0
-2
•
•
•
•
•
•
•
•
-5
•
•
•
1
•
•
•
•
•
•
•
•
•
5
0
2
•
•
•
•
•
•
•
•
•
•
10
15
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-1 0
-2
•
•
•
•
•
•
•
•
-5
•
•
•
1
•
•
•
•
•
•
•
•
•
5
0
2
•
•
•
•
•
•
•
•
•
•
10
15
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-1 0
-2
•
•
•
•
•
•
•
•
-5
•
•
•
1
2
•
••
•
•
•
•
••
••
•
•
•
•
5
0
•
•
•
•
10
15
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-1 0
-2
•
•
•
•
•
•
•
•
-5
•
•
•
1
2
•
••
•
•
•
•
••
••
•
•
•
•
5
0
•
•
•
•
10
15
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-1 0
-2
•
•
•
•
•
•
•
•
-5
•
•
•
1
2
•
•
•
•
••
••
•
•
•
•
••
•
•
10
15
•
5
0
•
•
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-1 0
-2
•
•
•
•
•
•
•
•
-5
•
•
•
1
2
•
•
•
•
• •
• •
•
•
•
•
• •
•
•
10
15
•
5
0
•
•
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-1 0
-2
•
•
•
•
•
•
•
•
-5
•
•
•
1
2
• •
• •
•
•
•
•
•
•
•
•
• •
•
•
10
15
•
5
0
•
•
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-1 0
-2
•
•
•
•
•
•
•
•
-5
•
•
•
1
2
• •
• •
•
•
•
•
•
•
•
•
• •
•
•
10
15
•
5
0
•
•
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-1 0
-2
•
•
•
•
•
•
•
•
-5
•
•
•
1
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
5
0
•
•
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-1 0
-2
•
•
•
•
•
•
•
•
-5
•
•
•
1
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-1 0
-2
•
•
•
•
•
•
•
•
-5
•
•
•
1
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-1 0
-2
•
•
•
•
•
•
•
•
-5
•
•
•
1
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-1 0
-2
•
•
•
•
•
•
•
•
-5
•
•
•
1
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-1 0
-2
•
•
•
•
•
•
•
•
-5
•
•
•
1
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-1 0
-2
•
•
•
•
•
•
•
•
-5
•
•
•
1
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-1 0
-2
•
•
•
•
•
•
•
•
-5
•
•
•
1
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
5
0
•
•
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-1 0
-2
•
•
•
•
•
•
•
•
-5
•
•
•
1
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-1 0
-2
•
•
•
•
•
•
•
•
-5
•
•
•
1
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-1 0
-2
•
•
•
•
•
•
•
•
-5
•
•
•
1
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-1 0
-2
•
•
•
•
•
•
•
•
-5
•
•
•
1
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-1 0
-2
•
•
•
•
•
•
•
•
-5
•
•
•
1
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-1 0
-2
•
•
•
•
•
•
•
•
-5
•
•
•
1
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-1 0
-2
•
••
•
•
••
••
•
-5
•
1
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-1 0
-2
•
••
•
•
••
••
•
-5
•
1
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-1 0
-2
•
•
••
••
•
•
•
••
-5
1
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-1 0
-2
•
•
• •
• •
•
•
•
• •
-5
1
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-1 0
-2
•
•
• •
• •
•
•
•
• •
-5
1
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-1 0
-2
•
•
• •
• •
•
•
•
• •
-5
1
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-1 0
-2
•
•
•
•
•
•
•
•
•
•
-5
•
1
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-1 0
-2
1
2
•
•
•
•
• •
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
-5
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-2
0
-1
1
2
•
•
•
•
• •
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
-5
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-2
0
-1
1
2
•
•
•
•
• •
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
-5
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-2
0
-1
1
2
•
•
•
•
••
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
-5
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-2
0
-1
1
2
•
•
•
•
••
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
-5
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-2
0
-1
1
2
•
•
•
•
••
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
-5
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-2
0
-1
1
2
•
•
•
•
••
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
-5
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-2
0
-1
1
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
-5
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-2
0
-1
1
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
-5
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-2
0
-1
1
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
-5
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-2
0
-1
1
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
-5
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-2
0
-1
1
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
-5
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-2
0
-1
1
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
-5
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-2
0
-1
1
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
-5
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-2
0
-1
1
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
-5
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-2
0
-1
1
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
-5
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-2
0
-1
1
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
-5
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-2
0
-1
1
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
-5
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-1 0
-2
1
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
-5
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-1 0
-2
1
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
-5
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-1 0
-2
1
2
•
•
•
•
• •
•
•
•
•
•
•
•
•
•
• •
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
-5
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-1 0
-2
•
•
•
•
•
•
•
• •
• •
•
•
•
-5
1
5
0
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-2
-1
•
•
•
•
•
•
•
• •
• •
•
•
•
-5
0
1
5
0
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-2
-1
•
•
•
•
•
•
•
••
••
•
•
•
-5
0
1
5
0
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
0
-1
-2
1
2
•
•
•
•
••
•
•
•
•
•
•
•
•
•
••
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
-5
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
0
-1
-2
1
2
•
•
•
•
••
•
•
•
•
•
•
•
•
•
••
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
-5
5
0
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
0
-1
-2
•
•
•
•
•
•
•
••
••
•
•
•
-5
1
5
0
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
•
•
•
•
•
•
0
-1
•
•
•
•
•
•
•
•
-2
-5
1
5
0
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
0
-2
•
•
•
•
•
•
-1
•
•
•
•
•
•
•
•
-5
1
5
0
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
0
-2
-1
•
•
•
•
•
•
•
•
•
•
•
•
•
•
-5
1
5
0
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
0
-2
•
•
•
•
•
•
-1
•
•
•
•
•
•
•
•
-5
1
5
0
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
•
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
0
-1
-2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
-5
1
5
0
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
•
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
0
-1
-2
•
•
•
•
•
•
••
•
•
•
•
•
•
-5
1
5
0
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
•
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
0
-1
-2
•
•
•
•
•
•
••
•
••
•
•
•
-5
1
5
0
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
•
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
0
-1
-2
•
•
•
•
•
•
••
•
••
•
•
•
-5
1
5
0
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
•
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-2
-1
•
•
•
•
•
•
••
•
••
•
•
•
-5
0
1
5
0
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
•
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-2
-1
•
•
•
•
•
•
• •
•
• •
•
•
•
-5
0
1
5
0
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-1 0
-2
•
•
•
•
•
•
• •
•
• •
•
•
•
-5
1
5
0
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
•
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-1 0
-2
•
•
•
•
•
•
•
• •
• •
•
•
•
-5
1
5
0
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
•
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Many second class particles plus one
̺
λ
-1 0
-2
•
•
•
•
•
•
•
•
•
•
•
•
-5
1
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
10
15
5
0
2
i
Couple three processes, and X (t) to Q(t).
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Microscopic convexity/concavity
We
say that a model has the microscopic convexity
property, if there is such a three-process coupling by which
Q(t) ≥ X (t)−tight error can be achieved.
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Microscopic convexity/concavity
We (almost) say that a model has the microscopic convexity
property, if there is such a three-process coupling by which
Q(t) ≥ X (t)−tight error can be achieved.
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Single Many
Microscopic convexity/concavity
We (almost) say that a model has the microscopic convexity
property, if there is such a three-process coupling by which
Q(t) ≥ X (t)−tight error can be achieved.
We (almost) say that a model has the microscopic concavity
property, if there is such a three-process coupling by which
Q(t) ≤ X (t)+tight error can be achieved.
C = H ′ (̺) = EQ/t
<
EX /t = R = [H(̺) − H(λ)]/(̺ − λ)
Models Hydrodynamics 2nd class Results Proof
Normal Abnormal
Normal fluctuations:
Once we have the microscopic convexity/concavity property,
Models Hydrodynamics 2nd class Results Proof
Normal Abnormal
Normal fluctuations:
Once we have the microscopic convexity/concavity property,
Theorem (Ferrari-Fontes (ASEP); B. (TAZRP, TABL))
Var(hVt (t))
= Var(ω) · |C − V |
t→∞
t
lim
h t
hVt (t)
t
Vt
0
0
t=0
i
Models Hydrodynamics 2nd class Results Proof
Normal Abnormal
Normal fluctuations:
Once we have the microscopic convexity/concavity property,
Theorem (Ferrari-Fontes (ASEP); B. (TAZRP, TABL))
Var(hVt (t))
= Var(ω) · |C − V |
t→∞
t
lim
h t
hVt (t)
t
Ct
Vt
0
0
t=0
i
Initial fluctuations are transported along the characteristics on
this scale.
Models Hydrodynamics 2nd class Results Proof
Normal Abnormal
Abnormal fluctuations:
Once we have the microscopic convexity/concavity property,
Models Hydrodynamics 2nd class Results Proof
Normal Abnormal
Abnormal fluctuations:
Once we have the microscopic convexity/concavity property,
On the characteristics V = C,
Theorem (B. - Komjáthy - Seppäläinen (ASEP, WASEP,
exponential concave TAZRP, exponential convex TABLP
so far...))
0 < lim inf
t→∞
Var(hCt (t))
Var(hCt (t))
≤ lim sup
< ∞.
t 2/3
t 2/3
t→∞
Models Hydrodynamics 2nd class Results Proof
Normal Abnormal
Abnormal fluctuations:
Once we have the microscopic convexity/concavity property,
On the characteristics V = C,
Theorem (B. - Komjáthy - Seppäläinen (ASEP, WASEP,
exponential concave TAZRP, exponential convex TABLP
so far...))
0 < lim inf
t→∞
Var(hCt (t))
Var(hCt (t))
≤ lim sup
< ∞.
t 2/3
t 2/3
t→∞
Important preliminaries were Cator and Groeneboom 2006, B.,
Cator and Seppäläinen 2006.
Models Hydrodynamics 2nd class Results Proof
Normal Abnormal
Abnormal fluctuations:
Once we have the microscopic convexity/concavity property,
On the characteristics V = C,
Theorem (B. - Komjáthy - Seppäläinen (ASEP, WASEP,
exponential concave TAZRP, exponential convex TABLP
so far...))
0 < lim inf
t→∞
Var(hCt (t))
Var(hCt (t))
≤ lim sup
< ∞.
t 2/3
t 2/3
t→∞
Important preliminaries were Cator and Groeneboom 2006, B.,
Cator and Seppäläinen 2006.
Other exclusion processes: Quastel and Valkó 2007.
Models Hydrodynamics 2nd class Results Proof
Normal Abnormal
Abnormal fluctuations:
Once we have the microscopic convexity/concavity property,
On the characteristics V = C,
Theorem (B. - Komjáthy - Seppäläinen (ASEP, WASEP,
exponential concave TAZRP, exponential convex TABLP
so far...))
0 < lim inf
t→∞
Var(hCt (t))
Var(hCt (t))
≤ lim sup
< ∞.
t 2/3
t 2/3
t→∞
Important preliminaries were Cator and Groeneboom 2006, B.,
Cator and Seppäläinen 2006.
Other exclusion processes: Quastel and Valkó 2007.
There is a huge literature now on limit distribution results, using
combinatorial and asymptotic analytic tools.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
λ
•
•
-5
•
•
•
0
5
•
•
10
15
Second class particle current: difference in growth.
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-1 0
-2
•
•
•
•
•
•
•
-5
0
1
2
•
•
•
•
•
•
•
•
•
10
15
5
•
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-1 0
-2
•
•
•
•
•
•
•
-5
0
1
2
•
•
•
•
•
•
•
•
•
10
15
5
•
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-1 0
-2
•
•
•
•
•
•
•
-5
0
1
2
•
•
•
•
•
•
•
•
•
10
15
5
•
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-1 0
-2
•
•
•
•
•
•
•
-5
0
1
2
•
•
•
•
•
•
•
•
•
10
15
5
•
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-1 0
-2
•
•
•
•
•
•
•
-5
0
1
•
•
•
•
5
•
•
•
•
2
•
•
•
•
10
15
Second class particle current: difference in growth.
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-1 0
-2
•
•
•
•
•
•
•
-5
0
1
•
•
•
•
5
•
•
•
•
2
•
•
•
•
10
15
Second class particle current: difference in growth.
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-1 0
-2
•
•
•
•
•
•
•
-5
0
1
2
•
••
•
•
••
•
•
10
15
5
•
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-1 0
-2
•
•
•
•
•
•
•
-5
0
1
2
•
• •
•
•
• •
•
•
10
15
5
•
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-1 0
-2
•
•
•
•
•
•
•
-5
0
1
2
•
• •
•
•
• •
•
•
10
15
5
•
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-1 0
-2
•
•
•
•
•
•
•
-5
0
1
2
•
• •
•
•
• •
•
•
10
15
5
•
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-1 0
-2
•
•
•
•
•
•
•
-5
0
1
•
•
•
•
•
•
•
•
•
10
15
5
•
2
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-1 0
-2
•
•
•
•
•
•
•
-5
0
1
•
•
•
•
•
•
•
•
•
10
15
5
•
2
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-1 0
-2
•
•
•
•
•
•
•
-5
0
1
•
•
•
•
•
•
•
•
•
10
15
5
•
2
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-1 0
-2
•
•
•
•
•
•
•
-5
0
1
•
•
•
•
•
•
•
•
•
10
15
5
•
2
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-1 0
-2
•
•
•
•
•
•
•
-5
0
1
•
•
•
•
•
•
•
•
•
10
15
5
•
2
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-1 0
-2
•
•
•
•
•
•
•
-5
0
1
•
•
•
•
•
•
•
•
•
10
15
5
•
2
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-1 0
-2
•
•
•
•
•
•
•
-5
0
1
•
•
•
•
•
•
•
•
•
10
15
5
•
2
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-1 0
-2
•
•
•
•
•
•
•
-5
0
1
•
•
•
•
•
•
•
•
•
10
15
5
•
2
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-1 0
-2
•
•
•
•
•
•
•
-5
0
1
•
•
•
•
•
•
•
•
•
10
15
5
•
2
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-1 0
-2
•
•
•
•
•
•
•
-5
0
1
•
•
•
•
•
•
•
•
•
10
15
5
•
2
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-1 0
-2
•
•
•
•
•
•
•
-5
0
1
•
•
•
•
•
•
•
•
•
10
15
5
•
2
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-1 0
-2
•
•
•
•
•
•
•
-5
0
1
•
•
•
•
•
•
•
•
•
10
15
5
•
2
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-1 0
-2
•
•
•
•
•
•
•
-5
0
1
•
•
•
•
•
•
•
•
•
10
15
5
•
2
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-1 0
-2
•
•
•
•
•
•
•
-5
0
1
•
•
•
•
•
•
•
•
•
10
15
5
•
2
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-1 0
-2
•
•
•
-5
•
•
•
•
0
1
•
•
•
•
•
•
•
•
•
10
15
5
•
2
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-1 0
-2
•
•
•
-5
•
•
•
•
0
1
•
•
•
•
•
•
•
•
•
10
15
5
•
2
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-1 0
-2
•
••
•
•
••
-5
0
1
•
•
•
•
•
•
•
•
•
10
15
5
•
2
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-1 0
-2
•
• •
•
•
• •
-5
0
1
•
•
•
•
•
•
•
•
•
10
15
5
•
2
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-1 0
-2
•
• •
•
•
• •
-5
0
1
•
•
•
•
•
•
•
•
•
10
15
5
•
2
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-1 0
-2
•
• •
•
•
• •
-5
0
1
•
•
•
•
•
•
•
•
•
10
15
5
•
2
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-1 0
-2
•
-5
•
•
•
•
•
•
0
1
•
•
•
•
•
•
•
•
•
10
15
5
•
2
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-1 0
-2
•
-5
•
•
•
•
•
1
• •
•
•
•
•
•
•
•
•
10
15
0
5
•
2
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-2
•
-5
0
-1
•
•
•
•
•
1
2
• •
•
•
•
•
•
•
•
•
10
15
0
5
•
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-2
•
-5
0
-1
•
•
•
•
•
1
2
• •
•
•
•
•
•
•
•
•
10
15
0
5
•
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-2
•
-5
0
-1
•
•
•
•
•
0
1
2
••
•
•
•
•
•
•
•
•
10
15
5
•
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-2
•
-5
0
-1
•
•
•
•
•
0
1
2
••
•
•
•
•
•
•
•
•
10
15
5
•
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-2
•
-5
0
-1
•
•
•
•
•
•
1
•
•
0
5
•
2
•
•
•
•
•
•
•
•
10
15
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-2
•
-5
0
-1
•
•
•
•
•
0
1
2
•
•
•
•
•
•
•
•
•
•
10
15
5
•
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-2
•
-5
0
-1
•
•
•
•
•
0
1
2
•
•
•
•
•
•
•
•
•
•
10
15
5
•
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-2
•
-5
0
-1
•
•
•
•
•
0
1
2
•
•
•
•
•
•
•
•
•
•
10
15
5
•
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-2
•
-5
0
-1
•
•
•
•
•
0
1
2
•
•
•
•
•
•
•
•
•
•
10
15
5
•
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-2
•
-5
0
-1
•
•
•
•
•
0
1
2
•
•
•
•
•
•
•
•
•
•
10
15
5
•
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-2
•
-5
0
-1
•
•
•
•
•
0
1
2
•
•
•
•
•
•
•
•
•
•
10
15
5
•
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-2
•
-5
0
-1
•
•
•
•
•
0
1
2
•
•
•
•
•
•
•
•
•
•
10
15
5
•
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-2
•
-5
0
-1
•
•
•
•
•
0
1
2
•
•
•
•
•
•
•
•
•
•
10
15
5
•
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-2
•
-5
0
-1
•
•
•
•
•
0
1
2
•
•
•
•
•
•
•
•
•
•
10
15
5
•
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-2
•
-5
0
-1
•
•
•
•
•
0
1
2
•
•
•
•
•
•
•
•
•
•
10
15
5
•
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-2
•
-5
0
-1
•
•
•
•
•
0
1
2
•
•
•
•
•
•
•
•
•
•
10
15
5
•
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-2
•
-5
0
-1
•
•
•
•
•
0
1
2
•
•
•
•
•
•
•
•
•
•
10
15
5
•
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-1 0
-2
•
-5
•
•
•
•
•
0
1
2
•
•
•
•
•
•
•
•
•
•
10
15
5
•
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-1 0
-2
•
-5
•
•
•
•
•
0
1
2
•
•
•
•
•
•
•
•
•
•
10
15
5
•
•
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-1 0
-2
•
-5
•
•
•
• •
•
•
•
0
1
5
•
2
•
•
•
•
•
•
•
•
10
15
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-1 0
-2
•
-5
•
•
•
• •
•
•
•
0
1
5
•
2
•
•
•
•
•
•
•
•
10
15
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-2
•
-5
-1
•
•
•
• •
•
•
•
0
0
1
5
•
2
•
•
•
•
•
•
•
•
10
15
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-2
•
-5
-1
•
•
•
••
•
•
•
0
0
1
5
•
2
•
•
•
•
•
•
•
•
10
15
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
0
-1
-2
•
-5
•
•
•
••
•
•
•
0
1
5
•
2
•
•
•
•
•
•
•
•
10
15
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
0
-1
-2
•
-5
•
•
•
•
•
1
•
•
•
0
5
•
2
•
•
•
•
•
•
•
•
10
15
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
0
-1
-2
•
-5
•
•
•
•
•
•
•
•
0
1
5
•
2
•
•
•
•
•
•
•
•
10
15
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
•
•
0
-1
•
•
•
•
•
•
-2
•
-5
0
1
5
•
2
•
•
•
•
•
•
•
•
10
15
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
0
-2
•
-5
•
•
-1
•
•
•
•
•
•
0
1
5
•
2
•
•
•
•
•
•
•
•
10
15
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
0
-2
•
-5
-1
•
•
•
•
•
•
•
•
0
1
5
•
2
•
•
•
•
•
•
•
•
10
15
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
0
-2
•
-5
•
•
-1
•
•
•
•
•
•
0
1
5
•
2
•
•
•
•
•
•
•
•
10
15
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
0
-1
-2
•
-5
•
•
••
•
•
•
•
0
1
5
•
2
•
•
•
•
•
•
•
•
10
15
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
0
-1
-2
•
-5
•
•
••
•
•
•
•
0
1
5
•
2
•
•
•
•
•
•
•
•
10
15
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
0
-1
-2
•
-5
•
•
••
•
•
•
•
0
1
5
•
2
•
•
•
•
•
•
•
•
10
15
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
0
-1
-2
•
-5
•
•
••
•
•
•
•
0
1
5
•
2
•
•
•
•
•
•
•
•
10
15
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-2
•
-5
-1
•
•
••
•
•
•
•
0
0
1
5
•
2
•
•
•
•
•
•
•
•
10
15
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-2
•
-5
-1
•
•
• •
•
•
•
•
0
0
1
5
•
2
•
•
•
•
•
•
•
•
10
15
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-1 0
-2
•
-5
•
•
• •
•
•
•
•
0
1
5
•
2
•
•
•
•
•
•
•
•
10
15
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-1 0
-2
•
-5
•
•
•
• •
•
•
•
0
1
5
•
2
•
•
•
•
•
•
•
•
10
15
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Proof: many second class particles
̺
λ
-1 0
-2
•
-5
•
•
•
•
•
•
•
0
1
•
5
•
2
•
•
•
•
•
•
•
•
10
15
Second class particle current: difference in growth.
•
i
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Upper bound (concave case)
P{Q(t) is too large}
Micro conc.: Q(t)<X (t)+ tight error
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Upper bound (concave case)
P{Q(t) is too large} ≤ P{X (t) is too large}
Micro conc.: Q(t)<X (t)+ tight error
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Upper bound (concave case)
P{Q(t) is too large} ≤ P{X (t) is too large}
≤ P{too many ’s have crossed Ct}
Micro conc.: Q(t)<X (t)+ tight error
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Upper bound (concave case)
P{Q(t) is too large} ≤ P{X (t) is too large}
≤ P{too many ’s have crossed Ct}
≤ P{hCt (t) − hCt (t) is too large}.
Micro conc.: Q(t)<X (t)+ tight error
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Upper bound (concave case)
P{Q(t) is too large} ≤ P{X (t) is too large}
≤ P{too many ’s have crossed Ct}
≤ P{hCt (t) − hCt (t) is too large}.
Centering hCt (t) − hCt (t) brings in a second-order
Taylor-expansion of H(̺).
Micro conc.: Q(t)<X (t)+ tight error
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Upper bound (concave case)
P{Q(t) is too large} ≤ P{X (t) is too large}
≤ P{too many ’s have crossed Ct}
≤ P{hCt (t) − hCt (t) is too large}.
Centering hCt (t) − hCt (t) brings in a second-order
Taylor-expansion of H(̺). This is another point where concavity of
the flux matters.
Micro conc.: Q(t)<X (t)+ tight error
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Upper bound (concave case)
P{Q(t) is too large} ≤ P{X (t) is too large}
≤ P{too many ’s have crossed Ct}
≤ P{hCt (t) − hCt (t) is too large(λ)}.
Centering hCt (t) − hCt (t) brings in a second-order
Taylor-expansion of H(̺). This is another point where concavity of
the flux matters.
Micro conc.: Q(t)<X (t)+ tight error
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Upper bound (concave case)
P{Q(t) is too large} ≤ P{X (t) is too large}
≤ P{too many ’s have crossed Ct}
≤ P{hCt (t) − hCt (t) is too large(λ)}.
Centering hCt (t) − hCt (t) brings in a second-order
Taylor-expansion of H(̺). This is another point where concavity of
the flux matters.
Optimize “too large(λ)” in λ,
Micro conc.: Q(t)<X (t)+ tight error
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Upper bound (concave case)
P{Q(t) is too large} ≤ P{X (t) is too large}
≤ P{too many ’s have crossed Ct}
≤ P{hCt (t) − hCt (t) is too large(λ)}.
Centering hCt (t) − hCt (t) brings in a second-order
Taylor-expansion of H(̺). This is another point where concavity of
the flux matters.
Optimize “too large(λ)” in λ, use Chebyshev’s inequality and
relate Var(hCt (t)) to Var(hCt (t)).
Micro conc.: Q(t)<X (t)+ tight error
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Upper bound (concave case)
P{Q(t) is too large} ≤ P{X (t) is too large}
≤ P{too many ’s have crossed Ct}
≤ P{hCt (t) − hCt (t) is too large(λ)}.
Centering hCt (t) − hCt (t) brings in a second-order
Taylor-expansion of H(̺). This is another point where concavity of
the flux matters.
Optimize “too large(λ)” in λ, use Chebyshev’s inequality and
relate Var(hCt (t)) to Var(hCt (t)).
The computations result in (remember E(Q(t)) = Ct )
P{Q(t) − Ct ≥ u} ≤ c ·
Micro conc.: Q(t)<X (t)+ tight error
t2
· Var(hCt (t)).
u4
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Upper bound (concave case)
P{Q(t) is too large} ≤ P{X (t) is too large}
≤ P{too many ’s have crossed Ct}
≤ P{hCt (t) − hCt (t) is too large(λ)}.
Centering hCt (t) − hCt (t) brings in a second-order
Taylor-expansion of H(̺). This is another point where concavity of
the flux matters.
Optimize “too large(λ)” in λ, use Chebyshev’s inequality and
relate Var(hCt (t)) to Var(hCt (t)).
The computations result in (remember E(Q(t)) = Ct )
P{Q(t) − Ct ≥ u} ≤ c ·
P{Q(t) − Ct ≥ u} ≤ c ·
t2
u4
t2
· Var(hCt (t)).
u4
· Var(hCt (t))
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Upper bound
Theorem (B. - Seppäläinen; also ideas from B. Tóth, H. Spohn
and M. Prähofer)
Started from (almost) equilibrium,
Var(hCt (t)) = c · E|Q(t) − C · t|
in the whole family of processes.
P{Q(t) − Ct ≥ u} ≤ c ·
t2
u4
· Var(hCt (t))
EQ(t) = Ct
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Upper bound
Theorem (B. - Seppäläinen; also ideas from B. Tóth, H. Spohn
and M. Prähofer)
Started from (almost) equilibrium,
Var(hCt (t)) = c · E|Q(t) − C · t|
in the whole family of processes.
Hence proceed with
P{Q(t) − Ct ≥ u} ≤ c ·
P{Q(t) − Ct ≥ u} ≤ c ·
t2
u4
t2
· E|Q(t) − C · t|
u4
· Var(hCt (t))
EQ(t) = Ct
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Upper bound
Theorem (B. - Seppäläinen; also ideas from B. Tóth, H. Spohn
and M. Prähofer)
Started from (almost) equilibrium,
Var(hCt (t)) = c · E|Q(t) − C · t|
in the whole family of processes.
Hence proceed with
t2
· E|Q(t) − C · t|
u4
t2
e
P{|Q(t)|
> u} ≤ c · 4 · E
u
P{Q(t) − Ct ≥ u} ≤ c ·
e
e
with Q(t)
: = Q(t) − Ct and E : = E|Q(t)|.
P{Q(t) − Ct ≥ u} ≤ c ·
t2
u4
· Var(hCt (t))
EQ(t) = Ct
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Upper bound
Theorem (B. - Seppäläinen; also ideas from B. Tóth, H. Spohn
and M. Prähofer)
Started from (almost) equilibrium,
Var(hCt (t)) = c · E|Q(t) − C · t|
in the whole family of processes.
Hence proceed with
t2
· E|Q(t) − C · t|
u4
t2
e
P{|Q(t)|
> u} ≤ c · 4 · E
u
P{Q(t) − Ct ≥ u} ≤ c ·
e
e
with Q(t)
: = Q(t) − Ct and E : = E|Q(t)|.
e
P{|Q(t)|
> u} ≤ c ·
t2
u4
·E
Models Hydrodynamics 2nd class Results Proof
Upper bound
e
We had P{|Q(t)|
> u} ≤ c ·
e
P{|Q(t)|
> u} ≤ c ·
t2
u4
·E
t2
u4
Upper bound Lower bound Micro c.
· E.
Models Hydrodynamics 2nd class Results Proof
Upper bound
Upper bound Lower bound Micro c.
2
e
We had P{|Q(t)|
> u} ≤ c · ut 4 · E.
Z ∞
e
e
P{|Q(t)|
> u} du
E = E|Q(t)| =
0
e
P{|Q(t)|
> u} ≤ c ·
t2
u4
·E
Models Hydrodynamics 2nd class Results Proof
Upper bound
Upper bound Lower bound Micro c.
2
e
We had P{|Q(t)|
> u} ≤ c · ut 4 · E.
Z ∞
e
e
P{|Q(t)|
> u} du
E = E|Q(t)| =
0
Z ∞
e
P{|Q(t)|
=E
> vE} dv
0
e
P{|Q(t)|
> u} ≤ c ·
t2
u4
·E
Models Hydrodynamics 2nd class Results Proof
Upper bound
Upper bound Lower bound Micro c.
2
e
We had P{|Q(t)|
> u} ≤ c · ut 4 · E.
Z ∞
e
e
P{|Q(t)|
> u} du
E = E|Q(t)| =
0
Z ∞
e
P{|Q(t)|
=E
> vE} dv
0
Z ∞
1
e
> vE} dv + E
P{|Q(t)|
≤E
2
1/2
e
P{|Q(t)|
> u} ≤ c ·
t2
u4
·E
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Upper bound
2
e
We had P{|Q(t)|
> u} ≤ c · ut 4 · E.
Z ∞
e
e
P{|Q(t)|
> u} du
E = E|Q(t)| =
0
Z ∞
e
P{|Q(t)|
=E
> vE} dv
0
Z ∞
1
e
> vE} dv + E
P{|Q(t)|
≤E
2
1/2
≤c·
that is, E 3 ≤ c · t 2 .
e
P{|Q(t)|
> u} ≤ c ·
t2
u4
·E
t2
1
+ E,
2
2
E
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Upper bound
2
e
We had P{|Q(t)|
> u} ≤ c · ut 4 · E.
Z ∞
e
e
P{|Q(t)|
> u} du
E = E|Q(t)| =
0
Z ∞
e
P{|Q(t)|
=E
> vE} dv
0
Z ∞
1
e
> vE} dv + E
P{|Q(t)|
≤E
2
1/2
≤c·
t2
1
+ E,
2
2
E
that is, E 3 ≤ c · t 2 .
Thm
Var(hCt (t)) = const. · E|Q(t) − Ct|
= const. · E ≤ c · t 2/3 .
e
P{|Q(t)|
> u} ≤ c ·
t2
u4
·E
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Lower bound
In the upper bound, the relevant orders were
u (deviation of Q(t)) ∼ t 2/3 ,
̺ − λ ∼ t −1/3 .
The lower bound works with similar arguments: compare
models of which the densities differ by t −1/3 , and use
connections between Q(t), X (t) and heights.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Lower bound
In the upper bound, the relevant orders were
u (deviation of Q(t)) ∼ t 2/3 ,
̺ − λ ∼ t −1/3 .
The lower bound works with similar arguments: compare
models of which the densities differ by t −1/3 , and use
connections between Q(t), X (t) and heights.
The critical feature in both the upper bound and lower bound
was microscopic convexity/concavity: Q(t) ≥ X (t) (convex) or
Q(t) ≤ X (t) (concave).
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Microscopic convexity/concavity
Model
H(̺) is
Micro c.?
t 2/3 law
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Microscopic convexity/concavity
Model
TASEP
H(̺) is
Micro c.?
t 2/3 law
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Microscopic convexity/concavity
Model
TASEP
H(̺) is
concave
Micro c.?
t 2/3 law
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Microscopic convexity/concavity
Model
TASEP
H(̺) is
concave
Micro c.?
Q(t) ≤ X (t)
t 2/3 law
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Microscopic convexity/concavity
Model
TASEP
H(̺) is
concave
Micro c.?
Q(t) ≤ X (t)
t 2/3 law
proved (B.-S.)
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Microscopic convexity/concavity
Model
TASEP
ASEP (WASEP)
H(̺) is
concave
Micro c.?
Q(t) ≤ X (t)
t 2/3 law
proved (B.-S.)
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Microscopic convexity/concavity
Model
TASEP
H(̺) is
concave
ASEP (WASEP)
concave
Micro c.?
Q(t) ≤ X (t)
t 2/3 law
proved (B.-S.)
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Microscopic convexity/concavity
Model
TASEP
H(̺) is
concave
Micro c.?
Q(t) ≤ X (t)
ASEP (WASEP)
concave
Q(t) ≤ X (t)+Err
t 2/3 law
proved (B.-S.)
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Microscopic convexity/concavity
Model
TASEP
H(̺) is
concave
Micro c.?
Q(t) ≤ X (t)
t 2/3 law
proved (B.-S.)
ASEP (WASEP)
concave
Q(t) ≤ X (t)+Err
proved (B.-S.)
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Microscopic convexity/concavity
Model
TASEP
H(̺) is
concave
Micro c.?
Q(t) ≤ X (t)
t 2/3 law
proved (B.-S.)
ASEP (WASEP)
concave
Q(t) ≤ X (t)+Err
proved (B.-S.)
rate 1 TAZRP
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Microscopic convexity/concavity
Model
TASEP
H(̺) is
concave
Micro c.?
Q(t) ≤ X (t)
t 2/3 law
proved (B.-S.)
ASEP (WASEP)
concave
Q(t) ≤ X (t)+Err
proved (B.-S.)
rate 1 TAZRP
concave
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Microscopic convexity/concavity
Model
TASEP
H(̺) is
concave
Micro c.?
Q(t) ≤ X (t)
t 2/3 law
proved (B.-S.)
ASEP (WASEP)
concave
Q(t) ≤ X (t)+Err
proved (B.-S.)
rate 1 TAZRP
concave
Q(t) ≤ X (t)
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Microscopic convexity/concavity
Model
TASEP
H(̺) is
concave
Micro c.?
Q(t) ≤ X (t)
t 2/3 law
proved (B.-S.)
ASEP (WASEP)
concave
Q(t) ≤ X (t)+Err
proved (B.-S.)
rate 1 TAZRP
concave
Q(t) ≤ X (t)
proved (B.-K.)
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Microscopic convexity/concavity
Model
TASEP
H(̺) is
concave
Micro c.?
Q(t) ≤ X (t)
t 2/3 law
proved (B.-S.)
ASEP (WASEP)
concave
Q(t) ≤ X (t)+Err
proved (B.-S.)
rate 1 TAZRP
concave
Q(t) ≤ X (t)
proved (B.-K.)
concave exp rate
TAZRP
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Microscopic convexity/concavity
Model
TASEP
H(̺) is
concave
Micro c.?
Q(t) ≤ X (t)
t 2/3 law
proved (B.-S.)
ASEP (WASEP)
concave
Q(t) ≤ X (t)+Err
proved (B.-S.)
rate 1 TAZRP
concave
Q(t) ≤ X (t)
proved (B.-K.)
concave exp rate
TAZRP
concave
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Microscopic convexity/concavity
Model
TASEP
H(̺) is
concave
Micro c.?
Q(t) ≤ X (t)
t 2/3 law
proved (B.-S.)
ASEP (WASEP)
concave
Q(t) ≤ X (t)+Err
proved (B.-S.)
rate 1 TAZRP
concave
Q(t) ≤ X (t)
proved (B.-K.)
concave exp rate
TAZRP
concave
Q(t) ≤ X (t)+Err
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Microscopic convexity/concavity
Model
TASEP
H(̺) is
concave
Micro c.?
Q(t) ≤ X (t)
t 2/3 law
proved (B.-S.)
ASEP (WASEP)
concave
Q(t) ≤ X (t)+Err
proved (B.-S.)
rate 1 TAZRP
concave
Q(t) ≤ X (t)
proved (B.-K.)
concave exp rate
TAZRP
concave
Q(t) ≤ X (t)+Err
proved (B.-K.-S.)
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Microscopic convexity/concavity
Model
TASEP
H(̺) is
concave
Micro c.?
Q(t) ≤ X (t)
t 2/3 law
proved (B.-S.)
ASEP (WASEP)
concave
Q(t) ≤ X (t)+Err
proved (B.-S.)
rate 1 TAZRP
concave
Q(t) ≤ X (t)
proved (B.-K.)
concave exp rate
TAZRP
concave
Q(t) ≤ X (t)+Err
proved (B.-K.-S.)
convex exp rate
TABLP
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Microscopic convexity/concavity
Model
TASEP
H(̺) is
concave
Micro c.?
Q(t) ≤ X (t)
t 2/3 law
proved (B.-S.)
ASEP (WASEP)
concave
Q(t) ≤ X (t)+Err
proved (B.-S.)
rate 1 TAZRP
concave
Q(t) ≤ X (t)
proved (B.-K.)
concave exp rate
TAZRP
concave
Q(t) ≤ X (t)+Err
proved (B.-K.-S.)
convex exp rate
TABLP
convex
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Microscopic convexity/concavity
Model
TASEP
H(̺) is
concave
Micro c.?
Q(t) ≤ X (t)
t 2/3 law
proved (B.-S.)
ASEP (WASEP)
concave
Q(t) ≤ X (t)+Err
proved (B.-S.)
rate 1 TAZRP
concave
Q(t) ≤ X (t)
proved (B.-K.)
concave exp rate
TAZRP
concave
Q(t) ≤ X (t)+Err
proved (B.-K.-S.)
convex exp rate
TABLP
convex
Q(t) ≥ X (t)−Err
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Microscopic convexity/concavity
Model
TASEP
H(̺) is
concave
Micro c.?
Q(t) ≤ X (t)
t 2/3 law
proved (B.-S.)
ASEP (WASEP)
concave
Q(t) ≤ X (t)+Err
proved (B.-S.)
rate 1 TAZRP
concave
Q(t) ≤ X (t)
proved (B.-K.)
concave exp rate
TAZRP
concave
Q(t) ≤ X (t)+Err
proved (B.-K.-S.)
convex exp rate
TABLP
convex
Q(t) ≥ X (t)−Err
proved (B.-K.-S.)
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Microscopic convexity/concavity
Model
TASEP
H(̺) is
concave
Micro c.?
Q(t) ≤ X (t)
t 2/3 law
proved (B.-S.)
ASEP (WASEP)
concave
Q(t) ≤ X (t)+Err
proved (B.-S.)
rate 1 TAZRP
concave
Q(t) ≤ X (t)
proved (B.-K.)
concave exp rate
TAZRP
concave
Q(t) ≤ X (t)+Err
proved (B.-K.-S.)
convex exp rate
TABLP
convex
Q(t) ≥ X (t)−Err
proved (B.-K.-S.)
less concave/convex
rate
(T)AZRP, (T)ABLP
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Microscopic convexity/concavity
Model
TASEP
H(̺) is
concave
Micro c.?
Q(t) ≤ X (t)
t 2/3 law
proved (B.-S.)
ASEP (WASEP)
concave
Q(t) ≤ X (t)+Err
proved (B.-S.)
rate 1 TAZRP
concave
Q(t) ≤ X (t)
proved (B.-K.)
concave exp rate
TAZRP
concave
Q(t) ≤ X (t)+Err
proved (B.-K.-S.)
convex exp rate
TABLP
convex
Q(t) ≥ X (t)−Err
proved (B.-K.-S.)
less concave/convex
rate
(T)AZRP, (T)ABLP
concave/
convex
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Microscopic convexity/concavity
Model
TASEP
H(̺) is
concave
Micro c.?
Q(t) ≤ X (t)
t 2/3 law
proved (B.-S.)
ASEP (WASEP)
concave
Q(t) ≤ X (t)+Err
proved (B.-S.)
rate 1 TAZRP
concave
Q(t) ≤ X (t)
proved (B.-K.)
concave exp rate
TAZRP
concave
Q(t) ≤ X (t)+Err
proved (B.-K.-S.)
convex exp rate
TABLP
convex
Q(t) ≥ X (t)−Err
proved (B.-K.-S.)
less concave/convex
rate
(T)AZRP, (T)ABLP
concave/
convex
??
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Microscopic convexity/concavity
Model
TASEP
H(̺) is
concave
Micro c.?
Q(t) ≤ X (t)
t 2/3 law
proved (B.-S.)
ASEP (WASEP)
concave
Q(t) ≤ X (t)+Err
proved (B.-S.)
rate 1 TAZRP
concave
Q(t) ≤ X (t)
proved (B.-K.)
concave exp rate
TAZRP
concave
Q(t) ≤ X (t)+Err
proved (B.-K.-S.)
convex exp rate
TABLP
convex
Q(t) ≥ X (t)−Err
proved (B.-K.-S.)
less concave/convex
rate
(T)AZRP, (T)ABLP
concave/
convex
??
??
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
̺
λ
-1 0
-2
•
•
-5
•
•
•
•
•
•
•
•
•
1
•
•
•
•
•
•
•
•
•
0
5
2
•
•
•
•
•
•
•
•
•
•
10
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
-2
-5
-1 0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1 0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1 0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1 0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1 0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1 0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1 0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1 0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1 0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1 0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1 0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1 0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1 0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1 0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1 0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1 0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1 0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1 0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1 0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1 0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1 0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1 0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1 0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1 0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1 0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1 0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1 0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1 0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1 0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1 0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1 0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1 0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
0
-1
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
0
-1
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
0
-1
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
0
-1
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
0
-1
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
0
-1
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
0
-1
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
0
-1
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
0
-1
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1
0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1
0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1
0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1
0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1
0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1
0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1
0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1
0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1 0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-1 0
-2
-5
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-1 0
-2
-5
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-1 0
-2
-5
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1
0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1
0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
0
-1
-2
-5
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
0
-1
-2
-5
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
0
-1
-2
-5
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
0
-1
-2
-5
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
0
-2
-5
-1
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = 0
−1
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
0
-2
-5
-1
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = −1
0
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
0
-2
-5
-1
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = −1
0
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
0
-1
-2
-5
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = −1
0
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
0
-1
-2
-5
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = −1
0
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
0
-1
-2
-5
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = −1
0
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
0
-1
-2
-5
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = −1
0
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1
0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = −1
0
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-2
-5
-1
0
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = −1
0
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-1 0
-2
-5
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = −1
0
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-1 0
-2
-5
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
mQ (t) = [the label of
at Q(t)] = −1
0
mQ (t) ≤ 0 ⇒ Q(t) ≤ X (t).
-1 0
-2
-5
0
1
5
10
2
15
i
Goal: to understand Q(t) on the background process of the ’s.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
This process mQ (t) is influenced by the background, and is
pretty complicated in general.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
This process mQ (t) is influenced by the background, and is
pretty complicated in general.
In the cases we succeeded so far, mQ (t) behaved nicely:
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
This process mQ (t) is influenced by the background, and is
pretty complicated in general.
In the cases we succeeded so far, mQ (t) behaved nicely:
◮
Either mQ (t) ≤ 0 a.s. (TASEP, Rate 1 TAZRP);
deterministicly adorable!
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
This process mQ (t) is influenced by the background, and is
pretty complicated in general.
In the cases we succeeded so far, mQ (t) behaved nicely:
◮
Either mQ (t) ≤ 0 a.s. (TASEP, Rate 1 TAZRP);
deterministicly adorable!
◮
Or mQ (t) ≤ Geometric (ASEP, concave exponential rate
d
d
TAZRP, ≥ −Geometric for convex exponential rate TABLP);
behaves like a drifted simple random walk.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
Q(t) ≤ X (t)+tight error
This process mQ (t) is influenced by the background, and is
pretty complicated in general.
In the cases we succeeded so far, mQ (t) behaved nicely:
◮
Either mQ (t) ≤ 0 a.s. (TASEP, Rate 1 TAZRP);
deterministicly adorable!
◮
Or mQ (t) ≤ Geometric (ASEP, concave exponential rate
d
d
TAZRP, ≥ −Geometric for convex exponential rate TABLP);
behaves like a drifted simple random walk.
This is the form of microscopic concavity we currently use:
mQ (t) is dominated by a time-independent distribution with
finite variance.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
The exponentially convex/concave rates make it possible to
separate the drift of mQ (t) from the background process: the
drift has a uniform lower bound for all background
configurations.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
The exponentially convex/concave rates make it possible to
separate the drift of mQ (t) from the background process: the
drift has a uniform lower bound for all background
configurations. Drifted random walk only wants to cross the
origin occasionally, hence the geometric bound.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
The exponentially convex/concave rates make it possible to
separate the drift of mQ (t) from the background process: the
drift has a uniform lower bound for all background
configurations. Drifted random walk only wants to cross the
origin occasionally, hence the geometric bound.
If we drop “exponentially”, we loose the uniform bound. Then
mQ (t) starts behaving like a diffusion.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
The exponentially convex/concave rates make it possible to
separate the drift of mQ (t) from the background process: the
drift has a uniform lower bound for all background
configurations. Drifted random walk only wants to cross the
origin occasionally, hence the geometric bound.
If we drop “exponentially”, we loose the uniform bound. Then
mQ (t) starts behaving like a diffusion. Diffusion in the random
environment of second class particles!
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
The exponentially convex/concave rates make it possible to
separate the drift of mQ (t) from the background process: the
drift has a uniform lower bound for all background
configurations. Drifted random walk only wants to cross the
origin occasionally, hence the geometric bound.
If we drop “exponentially”, we loose the uniform bound. Then
mQ (t) starts behaving like a diffusion. Diffusion in the random
environment of second class particles!
We don’t yet see the techniques to bound this diffusion in the
order of magnitude our arguments would require.
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
The critical feature: microscopic concavity
The exponentially convex/concave rates make it possible to
separate the drift of mQ (t) from the background process: the
drift has a uniform lower bound for all background
configurations. Drifted random walk only wants to cross the
origin occasionally, hence the geometric bound.
If we drop “exponentially”, we loose the uniform bound. Then
mQ (t) starts behaving like a diffusion. Diffusion in the random
environment of second class particles!
We don’t yet see the techniques to bound this diffusion in the
order of magnitude our arguments would require.
Once this is done, we could proceed with less and less
convex/concave models to see how t 1/3 scaling turns to t 1/4 for
linear models (random average process, linear rate AZRP)...
Models Hydrodynamics 2nd class Results Proof
Upper bound Lower bound Micro c.
Thank you.