Approach to equilibrium of diffusion
in a logarithmic potential
~t1/ 2
V (x)
~ t −β
~ t1/(b+1)
x
~ t −δ
x1(t)
x
Ori Hirschberg, David Mukamel, Gunter Schütz
Weizmann Institute of Science
LAFNES 11, Dresden, 11.7.11
Hirschberg, Mukamel, Schütz, arxiv:1106.0458 (2011)
Free diffusion
V (x)
x
∂
∂
P ( x, t ) = 2 P ( x, t )
∂t
∂x
2
Diffusion in a potential
V (x)
x
2

∂ 
∂
∂
P =  V ' ( x) + 2  P
∂x 
∂t
 ∂x
Diffusion in a potential
V (| x |>> 1)
~ b log x
x
2

∂
∂ b ∂ 
P≈
+ 2 P
∂t
 ∂x x ∂x 
Motivation
2

∂
∂ b ∂ 
P≈
+ 2 P
∂t
 ∂x x ∂x 
Appears in m a n y systems!
Motivation
Zero-range
Process
w(n) ~ 1 +
b
n
Atoms in optical lattices
w(4)
w(3)
w(2)
DNA denaturation
Diffusion in a potential:
Equilibrium solution
∂
∂
∂2 
P =  V ' ( x) + 2  P
∂t
∂x 
 ∂x
V ( x) ~ b log x
Diffusion in a potential:
Equilibrium solution
∂
∂
∂2 
P =  V ' ( x) + 2  P
∂t
∂x 
 ∂x
V ( x) ~ b log x
Boltzmann distribution:
*
P ( x) ~ e
−V ( x )
~x
−b
Diffusion in a potential:
Equilibrium solution
∂
∂
∂2 
P =  V ' ( x) + 2  P
∂t
∂x 
 ∂x
Boltzmann distribution:
V ( x) ~ b log x
*
P ( x) ~ e
−V ( x )
~x
−b
∞
b ≤1
*
P
∫ ( x)dx = ∞
−∞
No steady-state
Diffusion in a potential:
Equilibrium solution
∂
∂
∂2 
P =  V ' ( x) + 2  P
∂t
∂x 
 ∂x
Boltzmann distribution:
*
P ( x) ~ e
V ( x) ~ b log x
−V ( x )
~x
−b
∞
b ≤1
*
P
∫ ( x)dx = ∞
No steady-state
−∞
∞
b >1
∫ P ( x)dx = finite
*
−∞
Normalizable
steady-state
Diffusion in a potential:
Equilibrium solution
∂
∂
∂2 
P =  V ' ( x) + 2  P
∂t
∂x 
 ∂x
Boltzmann distribution:
*
P ( x) ~ e
V ( x) ~ b log x
−V ( x )
~x
−b
∞
b ≤1
*
P
∫ ( x)dx = ∞
No steady-state
−∞
∞
b >1
∫ P ( x)dx = finite
*
−∞
Normalizable
steady-state
The question
How does P(x,t) relax to equilibrium?
The question
How does P(x,t) relax to equilibrium?
P(x)
V (x)
x
The question
How does P(x,t) relax to equilibrium?
P(x)
V (x)
x
The question
How does P(x,t) relax to equilibrium?
P(x)
V (x)
x
The question
How does P(x,t) relax to equilibrium?
P(x)
V (x)
x
The question
How does P(x,t) relax to equilibrium?
P(x)
V (x)
P* ( x) ~ x − b
x
The question
How does P(x,t) relax to equilibrium?
The Answer
P ( x, t ) − P * ( x )
The question
How does P(x,t) relax to equilibrium?
The Answer
P ( x, t ) − P * ( x )
~ t1/ 2
~ t −β
~ t1/(b+1)
~ t −δ
x1(t)
x
Dynamics:
How to calculate?
Diffusion eq.
∂
∂2 
∂
P =  V ' ( x) + 2  P
∂t
∂x 
 ∂x
P( x, t ) = e −V ( x ) / 2ψ ( x, t )
Dynamics:
How to calculate?
Diffusion eq.
∂
∂2 
∂
P =  V ' ( x) + 2  P
∂t
∂x 
 ∂x
P( x, t ) = e −V ( x ) / 2ψ ( x, t )
Schrödinger eq.
with
∂ψ ∂ 2ψ
2
= 2 − VS ( x)ψ V ( x) = V ' ( x) − V ' ' ( x)
S
∂t
∂x
4
2
Dynamics:
How to calculate?
Diffusion eq.
∂
∂2 
∂
P =  V ' ( x) + 2  P
∂t
∂x 
 ∂x
P( x, t ) = e −V ( x ) / 2ψ ( x, t )
Schrödinger eq.
with
∂ψ ∂ 2ψ
2
= 2 − VS ( x)ψ V ( x) = V ' ( x) − V ' ' ( x)
S
∂t
∂x
4
2
ψ ( x, t ) = ∫ dEe a( E )ψ E ( x)
− Et
Dynamics in log potential:
How to calculate?
Diffusion eq.
2
∂
∂ b ∂ 
P≈
+ 2 P
∂t
 ∂x x ∂x 
V ( x) ~ b log x
P( x, t ) ~ x − b / 2ψ ( x, t )
(
∂ψ ∂ ψ
≈ 2 −
∂t
∂x
x
2
Schrödinger eq.
b b
+1
2 2
2
)ψ
ψ ( x, t ) = ∫ dEe a( E )ψ E ( x)
− Et
ψ E ~ Bessel function
Decay to equilibrium
in log potential
V ( x) ~ x 2
V ( x) = 0
“No ground state”
No gap
x
x2 ~ t
Ground state
Gap
x
x2 − x2
eq
~e
− Egap t
Decay to equilibrium
in log potential
V ( x) ~ x 2
V ( x) = 0
“No ground state”
No gap
x
x2 ~ t
Ground state
Gap
x
x2 − x2
eq
V ( x) ~ b log x
x
Ground state
No gap
x2 − x2
eq
~?
~e
− Egap t
Dynamics in log potential:
How to calculate?
Diffusion eq.
2
∂
∂ b ∂ 
P≈
+ 2 P
∂t
 ∂x x ∂x 
V ( x) ~ b log x
P( x, t ) ~ x − b / 2ψ ( x, t )
(
∂ψ ∂ ψ
≈ 2 −
∂t
∂x
x
2
Schrödinger eq.
b b
+1
2 2
2
)ψ
ψ ( x, t ) = ∫ dEe a( E )ψ E ( x)
− Et
ψ E ~ Bessel function
Dynamics in log potential:
How to calculate?
Diffusion eq.
2
∂
∂ b ∂ 
P≈
+ 2 P
∂t
 ∂x x ∂x 
V ( x) ~ b log x
P( x, t ) ~ x − b / 2ψ ( x, t )
(
∂ψ ∂ ψ
≈ 2 −
∂t
∂x
x
2
Schrödinger eq.
b b
+1
2 2
2
)ψ
ψ ( x, t ) = ∫ dEe a( E )ψ E ( x)
− Et
ψ E ~ Bessel function
Diffusion in log potential:
Scaling
P ( x, t ) − P ( x )
P * ( x)
*
x
Diffusion in log potential:
Scaling
P ( x, t ) − P ( x )
P * ( x)
*
x
at a given fixed t
Diffusion in log potential:
Scaling
t −δ g  x  for x ≤ x (t )
 1/( b +1) 
1
P ( x, t ) − P * ( x ) 


t
≈
*
P ( x)
 t − β f  1x/ 2  for x ≥ x1 (t )

t 
x
at a given fixed t
Diffusion in log potential:
Scaling
t −δ g  x  for x ≤ x (t )
 1/( b +1) 
1
P ( x, t ) − P * ( x ) 


t
≈
*
P ( x)
 t − β f  1x/ 2  for x ≥ x1 (t )

t 
~ t 1/ 2
~ t −β
x
at a given fixed t
Diffusion in log potential:
Scaling
t −δ g  x  for x ≤ x (t )
 1/( b +1) 
1
P ( x, t ) − P * ( x ) 


t
≈
*
P ( x)
 t − β f  1x/ 2  for x ≥ x1 (t )

t 
~ t 1/ 2
~ t −β
~ t 1/( b +1)
~t
−δ
x
at a given fixed t
Diffusion in log potential:
Scaling
t −δ g  x  for x ≤ x (t )
 1/( b +1) 
1
P ( x, t ) − P * ( x ) 


t
≈
*
P ( x)
 t − β f  1x/ 2  for x ≥ x1 (t )

t 
~ t 1/ 2
~ t −β
~ t 1/( b +1)
~t
−δ
Surprise!
x
at a given fixed t
Scaling at small x
P ( x , t ) − P * ( x ) −δ  x 
≈ t g γ 
*
P ( x)
t 
in
~ t1/(b+1)
~ t−δ
x
2
∂
∂ b ∂ 
P≈
+ 2 P
∂t
 ∂x x ∂x 
Scaling at small x
P ( x , t ) − P * ( x ) −δ  x 
≈ t g γ 
*
P ( x)
t 
in
2
∂
∂ b ∂ 
P≈
+ 2 P
∂t
 ∂x x ∂x 
b
g ' ' ( z ) + g ' ( z ) = −[γzg ' ( z ) + δg ( z )]t −(1− 2γ )
z
~ t1/(b+1)
~ t−δ
x
(z = x / t γ )
Scaling at small x
P ( x , t ) − P * ( x ) −δ  x 
≈ t g γ 
*
P ( x)
t 
in
2
∂
∂ b ∂ 
P≈
+ 2 P
∂t
 ∂x x ∂x 
0 for γ < 1 / 2
b
g ' ' ( z ) + g ' ( z ) = −[γzg ' ( z ) + δg ( z )]t −(1− 2γ )
z
~ t1/(b+1)
~ t−δ
x
(z = x / t γ )
Scaling at small x
P ( x , t ) − P * ( x ) −δ  x 
≈ t g γ 
*
P ( x)
t 
in
2
∂
∂ b ∂ 
P≈
+ 2 P
∂t
 ∂x x ∂x 
0 for γ < 1 / 2
b
g ' ' ( z ) + g ' ( z ) = −[γzg ' ( z ) + δg ( z )]t −(1− 2γ )
z
Solution
~ t1/(b+1)
~ t−δ
x
(z = x / t γ )
~
g ( z ) = C + Cz b +1
~
Find γ , δ , C , C
from continuity,
normalization.
Diffusion in log potential:
Scaling
t −δ g  x  for x ≤ x (t )
 1/( b +1) 
1
P ( x, t ) − P * ( x ) 


t
≈
*
P ( x)
 t − β f  1x/ 2  for x ≥ x1 (t )

t 
~ t 1/ 2
t 1/( b +1) << x1 (t ) << t 1/ 2
~ t −β
~ t 1/( b +1)
~t
−δ
x1 (t )
x
at a given fixed t
Diffusion in log potential:
Scaling
t −δ g  x  for x ≤ x (t )
 1/( b +1) 
1
P ( x, t ) − P * ( x ) 


t
≈
*
P ( x)
 t − β f  1x/ 2  for x ≥ x1 (t )

t 
~ t 1/ 2
~ t −β
x
at a given fixed t
What is β?
P ( x, t ) − P * ( x ) − β
≈t
*
P ( x)
x
f  1/ 2 
t 
in
~t1/ 2
~ t −β
x
2
∂
∂ b ∂ 
P≈
+ 2 P
∂t
 ∂x x ∂x 
What is β?
P ( x, t ) − P * ( x ) − β
≈t
*
P ( x)
x
f  1/ 2 
t 
in
2
∂
∂ b ∂ 
P≈
+ 2 P
∂t
 ∂x x ∂x 
u b

f ' ' (u ) +  −  f ' (u ) + β f (u ) = 0
2 u
~t1/ 2
~ t −β
x
(u = x / t 1/ 2 )
What is β?
P ( x, t ) − P * ( x ) − β
≈t
*
P ( x)
x
f  1/ 2 
t 
in
2
∂
∂ b ∂ 
P≈
+ 2 P
∂t
 ∂x x ∂x 
u b

f ' ' (u ) +  −  f ' (u ) + β f (u ) = 0
2 u
(u = x / t 1/ 2 )
Solution for every β!
~ t1/ 2
f (u ) = Cu
~ t −β
x
b +1
1
(
b +1
1 2
F
+ β;
b+3
u2
;−
2
4
)
Family of scaling solutions
β <1 ⇒
f (u )
f (u ) ~ u −2 β
−u 2 / 4
β =1 ⇒
f (u ) ~ e
β >1 ⇒
f (u ) ~ u −2 β with oscillations
f (u ) = Cu
b +1
1
(
b +1
1 2
F
+ β;
b+3
u2
;−
2
4
)
u
Surprise 2:
β selected by initial condition!
P ( x ,0 ) − P * ( x )
−a
~
x
P * ( x)
f (u)
β <1 ⇒
β =1 ⇒
β >1 ⇒
f (u ) ~ u −2 β
−u 2 / 4
f (u ) ~ e
f (u ) ~ u −2 β with oscillations
u
Surprise 2:
β selected by initial condition!
P ( x ,0 ) − P * ( x )
−a
~
x
P * ( x)
a
β=
2
f (u)
β <1 ⇒
β =1 ⇒
β >1 ⇒
f (u ) ~ u −2 β
−u 2 / 4
f (u ) ~ e
f (u ) ~ u −2 β with oscillations
u
Surprise 3:
β selected by initial condition!
P ( x ,0 ) − P * ( x )
−a
~
x
P * ( x)
a
β=
2
a
2
β =
1

f (u)
β <1 ⇒
β =1 ⇒
β >1 ⇒
f (u ) ~ u −2 β
−u 2 / 4
f (u ) ~ e
f (u ) ~ u −2 β with oscillations
u
a<2
a>2
Surprise 3:
Stability of solutions
From exact solution:
P ( x, t ) − P ( x) −1  x 
~ t f1  1 / 2 
*
P ( x)
t 
*
Localized
P ( x ,0 ) − P * ( x )
f (u)
β <1 ⇒
β =1 ⇒
β >1 ⇒
f (u ) ~ u −2 β
−u 2 / 4
f (u ) ~ e
f (u ) ~ u −2 β with oscillations
u
Surprise 3:
Stability of solutions
From exact solution:
P ( x, t ) − P ( x) −1  x 
~ t f1  1 / 2 
*
P ( x)
t 
*
Localized
P ( x ,0 ) − P * ( x )
If
P ( x, t 0 ) − P * ( x )
P * ( x)
=t
−β
0
 x 
f β  1/ 2  + δP
 t0 
Localized
perturbation
Surprise 3:
Stability of solutions
From exact solution:
P ( x, t ) − P ( x) −1  x 
~ t f1  1 / 2 
*
P ( x)
t 
*
Localized
P ( x ,0 ) − P * ( x )
If
P ( x, t 0 ) − P * ( x )
P * ( x)
=t
Then
P ( x, t ) − P * ( x )
P* ( x)
~ t −β
−β
0
 x 
Localized
f β  1/ 2  + δP
perturbation
t
 0 
 x  −1  x 
f β  1 / 2  + t f1  1 / 2 
t 
t 
Surprise 3:
Stability of solutions
From exact solution:
P ( x, t ) − P ( x) −1  x 
~ t f1  1 / 2 
*
P ( x)
t 
*
Localized
P ( x ,0 ) − P * ( x )
If
P ( x, t 0 ) − P * ( x )
P * ( x)
=t
Then
P ( x, t ) − P * ( x )
P* ( x)
~ t −β
−β
0
 x 
Localized
f β  1/ 2  + δP
perturbation
t
 0 
 x  −1  x 
f β  1 / 2  + t f1  1 / 2 
t 
t 
a / 2 a < 2
β =
a>2
 1
Selection in diffusion problem
is similar to
problems of propagating fronts
Propagation of fronts
F-KPP equation
Swift-Hohenberg eq.
Fisher, Kolmogorov, Petrovsky, Piscounoff
∂φ ∂ φ
= 2 + φ (1 − φ )
∂t ∂x
2
Hallatschek & Nelson (2009)
2
∂

∂φ
= εφ −  2 + 1 φ − φ 3
∂t
 ∂x

2
Fineberg & Steinberg (1987)
Propagation of fronts:
Velocity selection
φ (x)
F-KPP equation
∂φ ∂ 2φ
= 2 + φ (1 − φ )
∂t ∂x
Travelling wave solution
φ ( x, t ) ≈ Φ ( x − vt )
1
v
~ e − λx
v(λ ) λ < λ*
v= *
*
v
λ
>
λ

x
Diffusion vs. Front propagation
φ(x)
~ t 1/ 2
v
~ t −β
~ t1/(b+1)
~ t −δ
x1 (t )
2
∂
∂ b ∂ 
P≈
+ 2 P
∂t
 ∂x x ∂x 
x
x
∂φ ∂ 2φ
= 2 + φ (1 − φ )
∂t ∂x
Diffusion vs. Front propagation
φ(x)
~ t 1/ 2
v
~ t −β
~ t1/(b+1)
~ t −δ
x1 (t )
x
2
∂
∂ b ∂ 
P≈
+ 2 P
∂t
 ∂x x ∂x 
- Linear
- Depends on space
x
∂φ ∂ 2φ
= 2 + φ (1 − φ )
∂t ∂x
- Non-linear
- Translation invariant
Diffusion vs. Front propagation
φ(x)
~ t 1/ 2
v
~ t −β
~ t1/(b+1)
~ t −δ
x1 (t )
x
2
∂
∂ b ∂ 
P≈
+ 2 P
∂t
 ∂x x ∂x 
- Linear
- Depends on space
x
∂φ ∂ 2φ
= 2 + φ (1 − φ )
∂t ∂x
- Non-linear
- Translation invariant
- Selection depends on tail
- “Phase transition” in selection
- Oscillating solutions are unstable
What should you
remember?
What should you
remember?
~t1/2
~t
−β
~ t1/(b+1)
~t
−δ
x1(t)
x
a
2
β =
1

a<2
a>2
What should you
remember?
~t1/2
~t
−β
~ t1/(b+1)
~t
−δ
x1(t)
x
a
2
β =
1

a<2
a>2
1. Scaling at small x
2. Exponent β depends on initial
condition
3. “Phase transition” for β
Hirschberg, Mukamel, Schütz, arxiv:1106.0458 (2011)
What should you
remember?
~t1/2
~t
−β
~ t1/(b+1)
~t
−δ
x1(t)
x
a
2
β =
1

a<2
a>2
1. Scaling at small x
2. Exponent β depends on initial
condition
3. “Phase transition” for β
Hirschberg, Mukamel, Schütz, arxiv:1106.0458 (2011) Thank you!
From scaling to moving fronts
P ( x, t ) ~ t
−β
x

f α 
t 
ξ
x
=
e

Change variables: 
τ
t =e
P(ξ ,τ ) ~ e
− βτ
~
f (ξ − ατ )
Motivation:
Real-space condensation
For rates
w(n) ~ 1 +
b
+K
n
∂
P (n) = P (n + 1) w(n + 1) + P (n − 1) J − P (n)[ w( n) + J ] ≈
∂t
b
b

≈ P (n + 1)
− P(n) + P(n + 1) − 2 P( n) + P(n − 1)

n +1
n 
w( 4)
w(3)
w( 2)
Zero-range process
- Evans, Hanney (2005)
- Levine, Mukamel, Schütz (2005) …
Motivation:
Cold atoms in optical lattices
Laser “friction” force:
bp
f ( p) = −
1+ p2
b
V ( p) = log(1 + p 2 ) ~ b log p
2
Probability of momentum – semi-classical:
∂
∂ 
∂
W ( p, t ) =
− f ( p) +
W ( p, t )


∂x 
∂t
∂x 
- Castin, Dallibard, Cohen-Tannoudji (1991)
- Marksteiner, Ellinger, Zoller (1996)
- Lutz (2003) …
Motivation:
DNA denaturation
l
Entropy ~ log s l l −b
Energy
~ lε
0
∂ 
b  ∂2 
∂
P(l) ≈   (ε / T − s ) +  + 2  P(l)
∂t
l  ∂l 
 ∂l 
at T = Tc
- Poland, Scheraga (1966)
- Bar, Kafri, Mukamel (2007) …
Motivation:
Long-range interacting gases
Hamiltonian dynamics
2
N
1 N
pi
+
1 − cos(θ i − θ j )
H =∑
∑
2 N i , j =1
i =1 2
θi
[
]
Relaxation of a tagged particle
q = q( p)
- Bouchet, Dauxois (2005)
- Chavanis, Lemou (2007) …
∂
∂
∂2 
P(q) ≈  V ' (q) + 2  P(q)
∂t
∂q 
 ∂q
V (q ) = b log q
Initial distributions
P ( x ,0 ) − P * ( x )
−a
~x
*
P ( x)
P ( x , 0 ) = δ ( x − x0 )
P ( x,0) ~ P * ( x )[1 + Ax − a ]
a=0
A = −1 *
a<0
Leading term
a>0
Sub-leading term
* Levine, Mukamel, Schütz (2005)
Barkai, Kessler (2010)
a < 0 : Cold atoms experiment
1. Equilibrate system at b1
2. “Quench” to b2 > b1
3. Measure
p2
as function of t
4. Compare with theory, a = b1 – b2 <0
P ( x,0) − P * ( x) x − b1 − x − b2
− ( b1 − b2 )
~
x
=
P* ( x)
x −b2
a = 1 : Move potential
P(x)
P* ( x) ~ ( x − 1) − b
x
a = 1 : Move potential
P(x)
x
a = 1 : Move potential
P(x)
P ( x,0) ~ ( x − 1) −b ~ x −b (1 + bx −1 )
*
P ( x) ~ x
−b
x
a = 1 : Move potential
P(x)
P ( x,0) ~ ( x − 1) −b ~ x −b (1 + bx −1 )
*
P ( x) ~ x
−b
x
w( 4)
w(3)
w( 2)
Current correlations in ZRP
Zero-range process
Other a’s: absorbing boundary
Absorbing b.c. similar solution!
But, new definition of a
Important for questions of persistence &
first-passage properties
e.g., lifetime of DNA loop
Diffusion in log potential has
self-similarity of 2nd kind
V (x)
The limit
l→0
is singular!
x
When b<1
No singularity
V ( x) = b log x
l
Dimensional
analysis works
Diffusion in log potential:
Scaling
t −δ g  x  for x ≤ x (t )
 1/( b +1) 
1
P ( x, t ) − P * ( x ) 


t
≈
*
P ( x)
 t − β f  1x/ 2  for x ≥ x1 (t )

t 