Entropy production far from equilibrium in interacting many-body systems

Entropy production far from equilibrium in

interacting many-body systems

Sven Dorosz

University of Luxembourg

Far from Equilibrium

within a non equilibrium steady state (part I)

relaxation to a non equilibrium state

Table of contents

1

Models and Properties

2

Steady State Fluctuation Theorem

3

Rate Functions for Entropy Production

4

Analyzing Fluctuation Ratios for φ

5

_Conclusion

Stochastic Systems

M

1

M

2

′

M

_n

ASEP

A

+ A

_⇄

λ

h

A

+ 0

A

+ A

_⇄

λ

ε

λ

λ

A

+ 0

nA

_⇄

λ

ε

λ

λ

n0

0

⇄

h

ε

h

h

A

0

⇄

h

ε

h

h

A

Table:

The back-reactions are taking place with rates ε

h

h

and ε

λ

λ, with

Markov Dynamics and Master Equation

d

_P

_(C

_i

_{, t)}

d

_t

=

X

C

j

6=C

i

[ω(C

j

→ C

i

)P(C

j

, t) − ω(C

i

→ C

j

)P(C

i

, t)]

K

s

(C

i

, C

j

) = ω(C

j

→ C

i

)P

s

(C

j

) − ω(C

i

→ C

j

)P

s

(C

i

)

if K

s

(C

i

, C

j

) = 0 ∀i , j then detailed balance

Stationary State Probabilities

0 50 100 150 200 250 0 0.01 0.02 0.03 0.04

P

s

(C

i

)

0 50 100 150 200 250 0 0.005 0.01 0.015 0.02 0 50 100 150 200 250

configuration C

_i 0 0.005 0.01 0.015 0.02

P

s

(C

i

)

h=0.2 h=1.4 0 50 100 150 200 250

configuration C

_i 0 0.005 0.01 0.015 0.02 (a) (b) (c) (d)

Stationary State

K

=

X

i ,j

|ω(C

j

→ C

i

)P

s

(C

j

) − ω(C

i

→ C

j

)P

s

(C

i

)|

0 1 2 3 4

h

0 0.2 0.4 0.6 0.8

K

M1 M2’ M2 M3 0 0.2 0.4 0.6 0.8

ε

0 0.2 0.4 0.6 0.8

K

M1 M2’ M2 M3 ε=0 h=1

Stationary State

Steady State Fluctuation Theorem

All parameters are held fixed

Steady State Fluctuation Theorem

s

tot

= ln

Y

{i }

ω(C

i

→ C

i+1

)

ω(C

i+1

→ C

i

)

+ ln

P

s

(C

0

)

P

s

(C

τ

)

= s

m

+ ∆s

P(s

tot

)

P

(−s

tot

)

= exp(s

tot

)

Dissipation and Length Scales

Stationary State

-20 -10 0 10 20 30 40

s

_tot 0.0 0.5 1.0 1.5

p(s

tot

)

-20 -10 0 10 20

s

_tot -20 -10 0 10 20

ln[p(s

tot

)/p(-s

tot

)]

(a) (b)

Entropy Change

Numerical Method

L

µ

= −





X

j

ω(C

j

−→ C

i

)e

−µ ln

ω_ω(Cj →Ci ) (Ci →Cj )

_{− r (C}

_i

₎





= −





X

j

ω(C

j

−→ C

i

)

1−µ

ω(C

i

−→ C

j

)

µ

− r (C

i

)





Numerical Method

Properties of the LDF

ν(µ) = ν(1 − µ)

h˙s

m

i = dν(µ)/dµ|

_µ=0

Mean entropy production rates

0 0.5 1

α

0 2 4 6 8

<d

t

s

m

>

0 1 2 3 4 5

h

0 5 10 15

<d

t

s

m

>

0.2 0.4 0.6 0.8 α 0 100 200 300 - d 2 α <d t sm > (a) (b)

Numerical Method

Properties of the LDF

χ(σ) = max

µ

{ν(µ) − h˙s

m

iσµ}

χ(−σ) = χ(σ) + h˙s

m

iσ

Rate functions

-20 -10 0 10 20

σ

0 50 100

χ(σ)

-1.5 -1 -0.5 0 0.5 1 1.5

σ

-10 -5 0 5 10

χ

’

(σ)

(a) (b)

Rate functions

-1 0 1 2

µ

-0.2 -0.1 0.0 0.1

ν

(

µ)

-2 0 2 4

σ

0 0.5 1

χ

(

σ

)

-4 -2 0 2 4

σ

-0.5 0.0

χ

’(

σ

)

-4 -2 0 2 4

σ

0.00 0.25 0.50

χ

’’(

σ

)

-4 -2 0 2 4

σ

0.012 0.013 0.014

χ

’’(

σ)

(a) (b) (c) _(d) (e)

transient processes

the system is driven out of its steady state

by a time dependent reaction rate

The Jarzynski relation

System S described by a Hamiltonian H(λ)

h(t) = {h

0

, h

1

, ..., h

M−1

, h

M

}

definition of work W =

M−1

P

i

=0

H(C

i

, h

i+1

) − H(C

i

, h

i

)

he

−βW

i = e

−β∆F

∀ h(t)

The average is taken over all trajectories in configuration space.

The Crooks Relation

Comparing a process and its time reversed process:

P

F

(W ) = P

R

(−W )e

β(W −∆F )

D.E.Crooks, J.Stat. Phys. 90, 1481 (1998)

discrete case expressed as stationary probabilities

Three Reaction Diffusion Systems

M

1

M

2

′

M

₂

Configuration space

M_1, M_2’

λ

h

h

λ

D

D

D

n=5

n=4

λ

h

h

D

D

D

D

n=3

M_2

Exact Fluctuation Relation for φ

-4 -2 0 2 4 6 8

φ

-4 -2 0 2 4 6 8

ln (P

F

(

φ)/

P

R

(

−φ))

D=1 D=10 -5 0 5

φ

-5 0 5 10 h(t)~ sin(πt/(2M)) h(t)~ t3

(a)

(b)

Distributions for the Observable φ

-5 0 5

φ

10-4 10-2 100

P

F

(

φ),

P

R

(-φ)

-5 0 5

φ

10-4 10-2 100 -5 0 5

φ

10-4 10-2 100 -5 0 5

φ

10-4 10-2 100

P

F

(

φ),

P

R

(-φ)

-5 0 5

φ

10-4 10-2 100 (a) (b) (c) (d) (e)

General Fluctuation Ratios for φ

-6 -4 -2 0 2 4 6 8

φ

-8 -4 0 4 8

ln (P

F

(

φ)/

P

R

(

−φ))

D=5 D=10 -2 0 2 4 6

φ

-4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 8

φ

-3 -2 -1 0 1 2 3

ln (P

F

(

φ)/

P

R

(

−φ) − φ

-3 0 3 6

φ

-3 -2 -1 0 1 2 3 (c) (d) (a) (b)

Conclusion

Detailed characterization of non equilibrium steady states

(NESS) for reaction diffusion systems

Discussion of stationary probabilities

and |K | as a distance from equilibrium

Characterization of entropy production in NESS

Heterogeneous Nucleation near Structured Wall

Heterogeneous Nucleation near Structured Wall

Heterogeneous Nucleation near Structured Wall

Heterogeneous Nucleation