Large deviations
of the density
of the current
in non equilibrium
steadyandstates
Bernard DERRIDA
Université Pierre et Marie Curie
and
Ecole Normale Supérieure, Paris
June 2012
Lyon
COLLABORATORS
J.L. Lebowitz, E.R. Speer
C. Enaud
B. Douçot, P.-E. Roche
T. Bodineau
C. Appert, V. Lecomte, F. van Wijland
A. Gerschenfeld, E. Brunet
Outline
Density uctuations in non-equilibrium systems
Current uctuations
Open questions
Non steady state situations
Deterministic dynamics
Non equilibrium steady states: current of particles
Non equilibrium steady states: current of particles
ρa
System
ρb
Non equilibrium steady states: current of heat
T
a
System
T
b
Q
t
T
b
Ta
L
Equilibrium (Ta = Tb)
Steady state (Ta 6= Tb)
Q
t
T
b
Ta
L
Equilibrium (Ta = Tb)
( ) = P (−Q )
P Q
Steady state (Ta 6= Tb)
Q
t
T
b
Ta
L
Equilibrium (Ta = Tb)
( ) = P (−Q )
P Q
Steady state (Ta 6= Tb)
( ) ∼ P (−Q ) exp
P Q
Q
1
kTb
−
1
kTa
Fluctuation Theorem
Q
t
T
b
Ta
L
Equilibrium (Ta = Tb)
( ) = P (−Q )
P Q
Steady state (Ta 6= Tb)
( ) ∼ P (−Q ) exp
P Q
Q
1
kTb
−
1
kTa
Fluctuation Theorem
hQ 2 i
1
∼
t
L
hQ i
A(Ta , Tb )
∼
t
L
Fourier's law
Q
t
T
b
Ta
L
Equilibrium (Ta = Tb)
( ) = P (−Q )
P Q
Steady state (Ta 6= Tb)
( ) ∼ P (−Q ) exp
P Q
Q
1
kTb
−
1
kTa
Fluctuation Theorem
hQ 2 i
1
∼
t
L
hQ i
A(Ta , Tb )
∼
t
L
Fourier's law
( )?
P Q
Q
t
T
b
Ta
L
Equilibrium (Ta = Tb)
Non-equilibrium
(Ta 6= Tsteady
)b state
Q
t
T
b
Ta
L
Equilibrium (Ta = Tb)
P
(C) ∼ exp −
E
(C)
Non-equilibrium
(Ta 6= Tsteady
)b state
kT
Density uctuations
P (C) =?
Q
t
T
b
Ta
L
Equilibrium (Ta = Tb)
P
(C) ∼ exp −
E
(C)
Non-equilibrium
(Ta 6= Tsteady
)b state
P (C) =?
kT
Density uctuations
Short range correlations
Long range correlations
Q
t
T
b
Ta
L
Equilibrium (Ta = Tb)
P
(C) ∼ exp −
E
(C)
Non-equilibrium
(Ta 6= Tsteady
)b state
P (C) =?
kT
Density uctuations
Short range correlations
Time symmetry of uctuations
Long range correlations
Time asymmetry of uctuations
Density uctuations at equilibrium
Starting from S = k log Ω Einstein
N particles
n
N = Vρ
v
N
V
n particles in v
n particles in volume v
Density uctuations at equilibrium
Starting from S = k log Ω Einstein
N particles
n
N = Vρ
v
N
V
n particles in v
n particles in volume v
hn2 i − hni2 = kT v ρ κ
κ is the compressibility,
ρ the density in the reservoir
and T the temperature.
Density uctuations at equilibrium
Starting from S = k log Ω Einstein
N particles
n
N = Vρ
v
N
V
n particles in v
n particles in volume v
hn2 i − hni2 = kT v ρ κ
κ is the compressibility,
ρ the density in the reservoir
and T the temperature.
Variance of the uctuation = k × Response coecient
Boltzmann Constant k
∼ 10−23 J /K
Large deviations of the density
N particles
n
N = Vρ
v
N
Pro vn = r ∼ exp[−v a(r )]
n particles in v
V
a(r)
a(r ) is the large deviation function
ρ
r
At equilibrium a(r ) is the free energy
Large deviation functional
ρ1
ρ2
ρk
L
Pro(ρ1 , ...ρk ) ∼ exp −Ld F(ρ1 , ...ρk )
Large number k of boxes ~r = L~x
Pro({ρ(~x )}) ∼ exp −Ld F({ρ(~x )})
Exclusion processes
SSEP (Symmetric simple exclusion process)
1
α
1
1
β
1
δ
γ
1
L
ρa =
α
,
α+γ
ρb =
δ
β+δ
Large deviation function for the SSEP
Pro({ρ(x )}) ∼ exp[−LF({ρ(x )})]
ρa = ρb = F
Z 1
F({ρ(x )}) =
dx (1 − ρ(x )) log 1 1−−ρ(Fx ) + ρ(x ) log ρ(Fx )
0
Equilibrium
Large deviation function for the SSEP
Pro({ρ(x )}) ∼ exp[−LF({ρ(x )})]
ρa = ρb = F
Z 1
F({ρ(x )}) =
dx (1 − ρ(x )) log 1 1−−ρ(Fx ) + ρ(x ) log ρ(Fx )
0
Equilibrium
Non-equilibrium (ρa 6= ρb )
D Lebowitz Speer 2001-2002
Bertini De Sole Gabrielli Jona-Lasinio Landim 2002
F = sup
F (x )
Z
0
ρ(x )
F
(x )
1 − ρ(x )
dx (1 − ρ(x )) log 1 − F (x ) + ρ(x ) log F (x ) + log ρ − ρ
b
a
with F (x ) monotone, F (0) = ρa and F (1) = ρb
Non-equilibrium ρa 6= ρb
F = sup
F (x )
Z
0
ρ(x )
F
(x )
1 − ρ(x )
dx (1 − ρ(x )) log 1 − F (x ) + ρ(x ) log F (x ) + log ρ − ρ
b
a
⇓
F is non-local:
for example for small
F({ρ(x )}) =
+
Z
1
+ O (ρa − ρb )3
where
dx (1 − ρ(x )) log 11−−ρρ(∗ (xx))
0
2 Z 1
(ρa − ρb )
(ρa − ρ2a )2
ρa − ρb
0
dx
1
Z
x
+ ρ(x ) log
dy x (1 − y ) ρ(x ) − ρ∗ (x )
`
ρ(x )
ρ ∗ (x )
´`
´
ρ(y ) − ρ∗ (y )
ρ∗ (x ) = hρ(x )i =(1 − x )ρa + x ρb
Long-range correlations
Spohn 82
hρ(x )ρ(y )i − hρ(x )ihρ(y )i '
2
(ρa − ρb )
L G (x , y ) = − L x (1 − y )
1
Large deviation functional
Pro({ρ(x )}) ∼ exp[−Action] = exp [−LF({ρ(x )})]
Equilibrium
1. F local
R
2. F = T −1 d ~x f (ρ(~x ))
3. Short range correlations
f is the free energy per unit volume
Large deviation functional
Pro({ρ(x )}) ∼ exp[−Action] = exp [−LF({ρ(x )})]
Equilibrium
1. F local
R
2. F = T −1 d ~x f (ρ(~x ))
3. Short range correlations
f is the free energy per unit volume
Non-equilibrium
1. F non local
2. Weak long range correlations (SSEP for x < y )
h(ρ(x ) − ρ∗ (x ))(ρ(y ) − ρ∗ (y ))i '
1
Spohn 1982
2
(ρa − ρb )
G
(x , y ) = −
L
L x (1 − y )
3. Higher correlations
hρ(x1 )ρ(x2 ))...ρ(xn )ic ∼ L1−n
TWO APPROACHES
SSEP (Symmetric simple exclusion process)
1
α
1
1
β
1
δ
γ
1
L
Microscopic
Bethe ansatz, Perturbation theory, Computer,...
Macroscopic
i = Lx , t = L2 τ
"
Pro({ρ(x , τ ), j (x , τ )}) ∼ exp −L
Z
0
T /L2
dt
Z
0
1
[j + ρ0 ]2
dx
4ρ(1 − ρ)
#
Matrix ansatz
Fadeev 1980, ....,
D Evans Hakim Pasquier 1993
Steady state of the SSEP
1
α
1
1
β
1
δ
γ
1
L
W |X1 . . . XL |V i
P (τ1 , . . . , τL ) = hhW
|(D + E )L |V i
where
Xi =
(
D if site i occupied
E if site i empty
Matrix ansatz
Fadeev 1980, ....,
D Evans Hakim Pasquier 1993
Steady state of the SSEP
1
α
1
1
β
1
δ
γ
1
L
W |X1 . . . XL |V i
P (τ1 , . . . , τL ) = hhW
|(D + E )L |V i
where
Xi =
(
D if site i occupied
E if site i empty
hW |(αE − γ D ) = hW |
DE − ED = D + E
(β D − δ E )|V i = |V i
Large diusive systems
ρb
ρa
A diusive system
I
I
For ρa − ρb small:
ρa = ρb = ρ :
D (ρ) (ρa −ρb )
t =
L
hQt i
hQt2 i
σ(ρ)
t = L
Fick's law
Large diusive systems
ρb
ρa
A diusive system
I
I
For ρa − ρb small:
ρa = ρb = ρ :
D (ρ) (ρa −ρb )
t =
L
hQt i
Fick's law
hQt2 i
σ(ρ)
t = L
Note that σ(ρ) = 2kT ρ2 κ(ρ)D (ρ) where κ is the compressibility
Macroscopic uctuation theory
Onsager Machlup theory
for non equilibrium diusive systems
Kipnis Olla Varadhan 89
Spohn 91
..
.
Bertini De Sole Gabrielli
Jona-Lasinio Landim 2001 →
Evolution of a prole ρ(x , t ) for 0 ≤ t ≤ T
Pro({ρ(x , t ), j (x , t )})
"
exp −L
Z T /L2
0
dt
Z
0
1
dx
[j (x , t ) + ρ0 (x , t )D (ρ(x , t ))]2
2σ(ρ(x , t ))
I
d ρ = − dj (conservation law)
dt
dx
I
ρ(0, t ) = ρa ; ρ(1, t ) = ρb
#
Macroscopic uctuation theory
Pro({ρ(x , t ), j (x , t )})
"
exp −L
Z T /L2
0
dt
Z
0
1
dx
[j (x , t ) + ρ0 (x , t )D (ρ(x , t ))]2
2σ(ρ(x , t ))
m
j (x , t ) = −ρ0 (x , t )D (ρ(x , t )) + √1 η(x , t )
L
#
Macroscopic uctuation theory
Pro({ρ(x , t ), j (x , t )})
"
exp −L
Z T /L2
0
dt
Z
0
1
dx
[j (x , t ) + ρ0 (x , t )D (ρ(x , t ))]2
2σ(ρ(x , t ))
m
j (x , t ) = −ρ0 (x , t )D (ρ(x , t )) + √1 η(x , t )
L
with the white noise
hη(x , t )η(x 0 , t 0 )i = 2σ(ρ(x , t ))δ(x − x 0 )δ(t − t 0 )
#
dρ
dt
dj
= − dx
;
ρ(0, t ) = ρa ; ρ(1, t ) = ρb
Pro({ρ(x , t ), j (x , t )}) ∼ exp −Action
"
exp −L
Z T /L2
0
dt
Z
0
1
dx
=
[j (x , t ) + ρ0 (x , t )D (ρ(x , t ))]2
2σ(ρ(x , t ))
How does a uctuation ρ(x , 0) relax?
#
dρ
dt
dj
= − dx
ρ(0, t ) = ρa ; ρ(1, t ) = ρb
;
Pro({ρ(x , t ), j (x , t )}) ∼ exp −Action
"
exp −L
Z T /L2
0
dt
Z
0
1
dx
=
[j (x , t ) + ρ0 (x , t )D (ρ(x , t ))]2
2σ(ρ(x , t ))
How does a uctuation ρ(x , 0) relax?
Action = 0
⇔
dρ
dt
= (D (ρ(x , t ))ρ(x , t )0 )0
#
dρ
dt
dj
= − dx
ρ(0, t ) = ρa ; ρ(1, t ) = ρb
;
Pro({ρ(x , t ), j (x , t )}) ∼ exp −Action
"
exp −L
Z T /L2
0
dt
Z
0
1
dx
=
[j (x , t ) + ρ0 (x , t )D (ρ(x , t ))]2
2σ(ρ(x , t ))
How does a uctuation ρ(x , 0) relax?
Action = 0
⇔
dρ
dt
= (D (ρ(x , t ))ρ(x , t )0 )0
How is a uctuation ρ(x , 0) produced?
#
dρ
dt
dj
= − dx
ρ(0, t ) = ρa ; ρ(1, t ) = ρb
;
Pro({ρ(x , t ), j (x , t )}) ∼ exp −Action
"
exp −L
Z T /L2
0
dt
Z
0
1
dx
=
[j (x , t ) + ρ0 (x , t )D (ρ(x , t ))]2
2σ(ρ(x , t ))
How does a uctuation ρ(x , 0) relax?
Action = 0
⇔
dρ
dt
= (D (ρ(x , t ))ρ(x , t )0 )0
How is a uctuation ρ(x , 0) produced?
Minimize the Action with ρ(x , −∞) = ρsteady state (x )
#
Macroscopic uctuation theory
Bertini De Sole Gabrielli
Jona-Lasinio Landim 2001-2002
ρ (x)
ρa
ρ (x,t)
ρ*(x)
ρb
x
0
F({ρ(x )}) =
with
Z
min
ρ(x ,t )
0
−∞
dt
Z
0
1
1
(x , t )D (ρ(x , t ))]
dx [j (x , t ) +2ρσ(ρ(
x , t ))
ρ(x , 0) = ρ(x ) ; ρ(x , −∞) = ρ∗ (x )
0
2
SSEP
How does a uctuation ρ(x , 0) relax?
d ρ(x , t ) = d 2 ρ(x , t )
dt
dx 2
How is a uctuation ρ(x , 0) produced?
d ρ(x , s ) = d 2 ρ(x , s ) −2 (ρa − ρb ) (1−2ρ(x , s )) d ρ(x , s ) +O
ds
dx 2
ρa (1 − ρa )
dx
(ρa − ρb )2
Current uctuations in non-equilibrium steady states
Ta
Tb
Current uctuations in non-equilibrium steady states
Ta
Current uctuations on a ring
T
Tb
AVERAGE CURRENT OF HEAT
Q
t
T
b
Ta
L
hQt i
= F (Ta , Tb , L, contacts)
t
AVERAGE CURRENT OF HEAT
Q
t
T
b
Ta
hQt i
= F (Ta , Tb , L, contacts)
t
L
FOURIER's law:
large L
large L and Ta − Tb small
hQt i
G (Ta , Tb )
'
t
L
hQt i
Ta − Tb
' D (T )
' −D (T )∇T
t
L
AVERAGE CURRENT OF PARTICLES
Q
t
ρ
ρa
L
b
hQt i
= F (ρa , ρb , L, contacts)
t
AVERAGE CURRENT OF PARTICLES
Q
t
ρ
ρa
b
hQt i
= F (ρa , ρb , L, contacts)
t
L
FICK's law:
large L
large L and ρa − ρb small
hQt i
G (ρa , ρb )
'
t
L
hQt i
ρa − ρb
' D (ρ)
' −D (ρ)∇ρ
t
L
ONE DIMENSIONAL MECHANICAL SYSTEMS
Ideal gas
No Fourier's law
Harmonic chain
E=
X
pi2 + X g (x − x )2
i +1
i
2m
i
hQt i
t
' G (Ta , Tb )
ONE DIMENSIONAL MECHANICAL SYSTEMS
Ideal gas
No Fourier's law
Harmonic chain
E=
X
pi2 + X g (x − x )2
i +1
i
2m
i
hQt i
t
' G (Ta , Tb )
Hard particle gas
Anomalous Fourier's law
Ta
Anharmonic chain
Ta
hQt i
t
.3 ≤ α ≤ .5
'
G (Ta , Tb )
L1−α
Delni Lepri Livi Politi Livi,
Spohn, Grassberger, ...
DIFFUSIVE SYSTEMS: average current
Symmetric exclusion
Fourier's law
1
α
1
1
β
1
δ
γ
1
L
General lattice gas
Random walkers
KMP model
Kipnis Marchioro Presutti
hQt i
t
'
G (Ta , Tb )
L
SSEP (Symmetric simple exclusion process)
D. Douçot Roche 2004
1
α
1
1
β
1
δ
γ
1
L
ρa =
lim
t →∞
hQ (t )i
t
'
α
,
α+γ
1
L [ρa − ρb ]
hQ 2 (t )ic
1
t
L
ρb =
δ
β+δ
Fick0 s law
2(ρ2a + ρa ρb + ρ2b )
lim
t →∞
t ' L ρa + ρb −
3
3
16ρ2a + 28ρa ρb + 16ρ2b
hQ (t )ic
1
lim
' (ρa − ρb ) 1 − 2(ρa + ρb ) +
t →∞
15
CURRENT FLUCTUATIONS AND LARGE DEVIATIONS
Q
Tb
Ta
Qt Energy transferred during time t
Pro Qtt
=j
∼ exp[−t
F(j)
F (j )]
Expansion of F (j ) near j gives all cumulants of Qt
∗
j*
j
DIFFUSIVE SYSTEMS: higher cumulants
Q
t
ρ
ρa
b
Bodineau D. 2004
L
I
I
For ρa − ρb small:
ρa = ρb = ρ :
hQt i
D (ρ)(ρa −ρb )
t =
L
hQt2 i
σ(ρ)
t = L
Then one can calculate
All cumulants of Qt for arbitrary ρa and ρb
ADDITIVITY PRINCIPLE
Pro Qtt
= j , ρa , ρb
Q
ρ
∼ exp[−t FL+L0 (j , ρa , ρb )]
Bodineau D. 2004
t
ρ
a
L
b
L’
Additivity
PL+L (Q , ρa , ρb ) ∼ max
[PL (Q , ρa , ρ) PL (Q , ρ, ρb )]
ρ
0
[FL (j , ρa , ρ)
FL+L (j , ρa , ρb ) = min
ρ
0
0
+
FL (j , ρ, ρb )]
0
Bodineau D. 2004
VARIATIONAL PRINCIPLE
FL (j , ρa , ρb ) = L1 min ρ(x )
1
Z
0
dx [j L
+ ρ0 (x ) D (ρ(x ))]2
2σ(ρ(x ))
ρ (x)
ρa
ρ* (x)
ρ
x
0
Satises the uctuation theorem
F (j ) − F (−j ) = j
Z
ρb
ρa
D (ρ) d ρ
σ(ρ)
b
1
Gallavotti Cohen
Evans Searls
.... Kurchan
Lebowitz Spohn
1995
1994
1998
1999
DIFFUSIVE SYSTEMS: all the cumulants
Bodineau D. 2004
hQt ic
t
hQt2 ic
t
hQt3 ic
t
hQt4 ic
t
=
=
=
=
1
L I1
1 I2
L
I1
1 3 ( I3 I1 − I22 )
L
I13
1 3 ( 5 I4 I12 − 14 I1 I2 I3 + 9 I23 )
L
I15
where
In =
For the SSEP D (ρ) = 1 and σ(ρ) = 2ρ(1 − ρ)
For the KMP D (ρ) = 1 and σ(ρ) = 4ρ2
For the random walkers D (ρ) = 1 and σ(ρ) = 2ρ
Z
ρa
ρb
D (ρ)σ(ρ)n−1 d ρ
TRUE VARIATIONAL PRINCIPLE
Bertini De Sole Gabrielli
Jona-Lasinio Landim 2005
1
F (j ) = L1 Tlim
min
→∞ ρ(x ,t ),j (x ,t ) T
with
dρ
dt
Z T
0
dt
1
Z
0
(x , t )D (ρ(x , t ))]
dx [j (x , t ) +2ρσ(ρ(
x , t ))
0
2
dj
= − dx
(conservation), ρt (0) = ρa , ρt (1) = ρb and
jT=
Z T
0
jt (x )dt
I
Sucient condition for the optimal prole to be time independent
I
Dynamical phase transition
the optimal
ρ t (x )
Bodineau D. 2005-2007
starts to become time dependent
SSEP ON A RING
Appert D Lecomte Van Wijland 2008
N particles
L sites
1
1
ρ=
N
L
Qt ux through a
bond during time t
CUMULANTS OF THE CURRENT FOR THE SSEP ON A RING
particles and L sites
σ(ρ) = 2ρ(1 − ρ) = 2N (LL2−N )
hQ 2 ic
t
hQ 4 ic
t
N
σ
= L−
1
2
= 2(Lσ−1)2
hQ 6 ic
2
3
(L−1)σ 2
= − (L −4L(+L2−)σ1)3−(L2−
2)
hQ 8 ic
4
3
2
18)σ 4 −4(L−1)(11L2 −L+12)σ 3 +48(L−1)2 σ 2
= (10L −2L +27L −15L+24
(L−1)4 (L−2)(L−3)
t
t
hQ 2 ic
σ
=
Gaussian +
t
L
hQ 2n ic
σn
∼ 2 for n ≥ 2
t
L
Fick's law
UNIVERSAL CUMULANTS OF THE CURRENT
hQ 2 ic
σ
=
t
L
Gaussian
hQ 4 ic
hQ 6 ic
hQ 8 ic
σ2
σ3
5σ 4
' 2,
' − 2,
'
t
2L
t
4L
t
12L2
Universal
UNIVERSAL CUMULANTS OF THE CURRENT
hQ 2 ic
σ
=
t
L
Gaussian
hQ 4 ic
hQ 6 ic
hQ 8 ic
σ2
σ3
5σ 4
' 2,
' − 2,
'
t
2L
t
4L
t
12L2
Macroscopic uctuation theory
Universal
UNIVERSAL CUMULANTS OF THE CURRENT
hQ 2 ic
σ
=
t
L
Gaussian
hQ 4 ic
hQ 6 ic
hQ 8 ic
σ2
σ3
5σ 4
' 2,
' − 2,
'
t
2L
t
4L
t
12L2
Macroscopic uctuation theory
+ uctuations around a at prole
Universal
DIFFUSIVE SYSTEMS: summary
Open system
Ta
Tb
hQ n ic /t ∼ L−1
Ring
T
hQ 2 ic /t ∼ L−1
hQ 2n ic /t ∼ L−2
for
n≥2
HARD PARTICLE GAS: the second cumulant
Ta
Ta
HARD PARTICLE GAS: the second cumulant
Ta
Ta
0.45
2000 �Q2t �ring
�Q2t �ring /t
2
2
�Qt �ring
�Qt �ring /t
0.4
1500
0.35
1000
0.3
0.25
500
0.2
a)
0
2000
4000
t
6000
hQ 2 i
t
b)
0
has a limit
2 · 10−4
1/t
4 · 10−4
HARD PARTICLE GAS: Size dependence of the cumulants
OPEN SYSTEM
t→∞
lim �Qnt �c /t
t→∞
101
100
10−1
101
100
10−1
T a > Tb
Ta = Tb
102
a)
RING
103
N
lim �Qnt �c /t
�Q4t �ring
c
102 �Q2t �ring
c
t→∞
�Q4t �open
c
�Q2t �open
c
102
lim �Qnt �c /t
�Q4t �open
c
�Q3t �open
c
�Q2t �open
c
�Qt �open
102
101
100
10−1
102
103
N
b)
102
103
N
HARD PARTICLE GAS: Size dependence of the cumulants
OPEN SYSTEM
t→∞
lim �Qnt �c /t
t→∞
101
100
10−1
101
100
10−1
T a > Tb
Ta = Tb
102
a)
RING
�Q4t �open
c
�Q2t �open
c
102
lim �Qnt �c /t
�Q4t �open
c
�Q3t �open
c
�Q2t �open
c
�Qt �open
102
103
N
b)
102
103
N
t→∞
lim �Qnt �c /t
�Q4t �ring
c
102 �Q2t �ring
c
101
ANOMALOUS FOURIER'S LAW
100
10−1
102
103
N
Correlations on a mesoscopic scale
Ta
Ta
L Cor(x,.25)
1
0
−2
100
200
−4
−6
400
0
0.2
0.4
0.6
0.8
1
x
Delni, Lepri, Livi, Mejia-Monasterio, Politi 2010
Gerschenfeld, D., Lebowitz 2010
Step initial condition
Q
−1
0
ASEP
t
1
G. Schütz 98
..
.
Prähofer Spohn 2000-2002
..
.
Tracy Widom 2008
2
Qt
ρa
ρb
SSEP
D Gerschenfeld 2009
ρa
ρb
λQ e t =?
Step initial condition on the innite line
SSEP
ρ
Qt
a
D Gerschenfeld 2009
ρb
λQ e t '
with H (ω) =
and
1
π
R∞
∞
exp[√t H (ω)]
h
i
2
dk log 1 + ω e −k
ω = 1 − [1 − (e λ − 1)ρa ][1 − (1 − e −λ )ρb ]
Step initial condition on the innite line
SSEP
ρ
Qt
a
D Gerschenfeld 2009
ρb
λQ e t '
with H (ω) =
and
1
π
R∞
∞
exp[√t H (ω)]
h
i
2
dk log 1 + ω e −k
ω = 1 − [1 − (e λ − 1)ρa ][1 − (1 − e −λ )ρb ]
π2 3
For large Qt : Pro(Qt ) ∼ exp −
Q /t
12 t
Density proles condionned on the current
Density
0.8
0
t/2
0.6
0.4
t
0.2
0
space
Density proles condionned on the current
Density
0.8
0
t/2
0.6
0.4
t
0.2
0
space
OPEN QUESTIONS
Large deviation function of the density
for a general diusive system
Non steady state situations
Theory for mechanical systems which conserved impulsion
Mean eld theory for large deviation functions