entropy production and fluctuation phenomena in nonequilibrium

Entropy production and fluctuation phenomena in nonequilibrium systems

Haye HinrichsenFaculty for Physics and AstronomyUniversity of Würzburg, Germany

Workshop on Large Fluctuations in Non­Equilibrium SystemsMPIPKS Dresden, July 2011

In collaboration with:

Andre Barato, ICTP, Trieste, ItalyUrna Basu, SAHA Institute, Kolkata, IndiaRaphael Chetrite, Lyon and CNRSChristian Gogolin, PotsdamPeter Janotta, WürzburgDavid Mukamel, Weizmann Insititute, Israel

Non-equilibrium Dynamics, Thermalization and Entropy ProductionH. Hinrichsen, C. Gogolin, and P. JanottaJ. Phys.: Conf. Ser. 297 012011 (2011)

Entropy production and fluctuation relations for a KPZ interfaceA. C. Barato, R. Chetrite, H. Hinrichsen, and D. MukamelJ. Stat. Mech.: Theor. Exp. P10008 (2010)

Outline

1) Introduction to entropy production

2) Fluctuation theorem revisited

3) Entropy production and renormalization

Nonequilibrium systemsNonequilibrium systems

T1 T2

μ1 μ2

Flow of heat

… typically driven systems

Flow of particles

Environment

Nonequilibrium systemsNonequilibrium systems

System

Environment

Nonequilibrium systemsNonequilibrium systems

System

drive

entropy

Models of classical nonequilibrium systemsModels of classical nonequilibrium systems

Systementropy

Model

Models of classical nonequilibrium systemsModels of classical nonequilibrium systems

Systementropy

Set of configurations Ωsys

(state space)

configurations c∈ sys

Model

Models of classical nonequilibrium systemsModels of classical nonequilibrium systems

Systementropy

Model Irreversible dynamics byspontaneous transitions 

at ratecc ' wcc '

Ωsys

Configurational entropy 

Environmental entropy            

Total entropy 

S sys t = −ln P c , t

S env t

S tot (t)=Ssys(t )+Senv(t )

EnvironmentSystem

drive

entropy

Actual time evolution:

Sequence of transitions(stochastic path)

at times                        

Our partial knowledge:

Probability distribution P(c,t)evolving deterministicallyby the master equation.

c1c2c3 ...cN

t1 , t 2 , t3 , ... , t N

⟨Ssys(t)⟩ = −∑c∈Ωsys

P(c , t) ln P(c , t)

Ssys(t) = −ln P (c(t), t)

ddt

P (c , t) = ∑c '∈Ω

P(c ' , t)wc ' c−P (c ,t )w cc '

Configurational entropy 

Mean entropy

Entropy of the systemEntropy of the system

⟨Ssys(t)⟩ = −∑c∈Ωsys

P(c , t) ln P(c , t)

Ssys(t) = −ln P (c(t), t)Configurational entropy 

Mean entropy

Entropy of the systemEntropy of the system

Change of conf. entropy  S sys(t) = −P (c (t), t)P (c (t), t)

−∑j

δ(t−t j) lnP (c j , t )

P (c j−1 , t )

⟨ Ssys(t)⟩ = − ∑c , c '∈Ωsys

P (c ,t )wc→c ' lnP (c ,t )P(c ' , t)

Change of mean entropy

Configurational entropy 

Environmental entropy            

Total entropy 

S sys t = −ln P c , t

S env t

S tot (t)=Ssys(t )+Senv(t )

EnvironmentSystem

drive

entropy

??

Senv(t) = ∑j

δ(t−t j) lnωc j−1→ c j

ωc j→c j−1

Andrieux and Gaspard, J. Chem. Phys. 2004U. Seifert, PRL 2005

Commonly accepted formula for theCommonly accepted formula for theenvironmental entropyenvironmental entropy

Where does it come from?

1976

X 1 X 2 X N

P(c , t)

[X i](t )

probability

concentration

Schnakenberg: The master equation

is mapped to a fictitious chemical system evolving according to the law of mass action (= mean field equation)

Fictitious chemical systemFictitious chemical system

ddt

P(c , t) = ∑c '∈Ω

(P(c ' , t)wc '→c−P(c , t)wc→c ')

Isothermal / isochroric   → minimize F.

Extent of reactionExtent of reaction = average number of forward reactions c→c' minus backward reactions c'→c.

Brief summary of Schnakenbergs argument (1)Brief summary of Schnakenbergs argument (1)

Thermodynamic fluxThermodynamic flux Conjugate thermodynamic forceConjugate thermodynamic force

Extent of reaction Extent of reaction ξξcc′cc′

Chemical a nityffiChemical a nityffi

Compare

with

 → Chemical affinity is chemical potential difference

With  and   

we arrive at:

Brief summary of Schnakenbergs argument (2)Brief summary of Schnakenbergs argument (2)

F = ∑cc '

Acc '˙ξcc ' = −∑

cc '

A cc '˙N cc '

Brief summary of Schnakenbergs argument (3)Brief summary of Schnakenbergs argument (3)

In the stationary state                             we have  

With                           . Hence                             turns intoF=∑cc '

Acc '˙ξcc '

E,T constant

Brief summary of Schnakenbergs argument (4)Brief summary of Schnakenbergs argument (4)

S=−kB∑c , c '

ξc , c ' ln[X c ' ]wc '→c

[X c ]wc→c '

S = −kB∑c , c '

ξc , c ' ln[X c ' ]

[X c]− kB∑

c , c '

ξc , c ' lnwc '→c

wc→c '

Brief summary of Schnakenbergs argument (4)Brief summary of Schnakenbergs argument (4)

⟨ Ssys(t)⟩ = − ∑c , c '∈Ωsys

P (c ,t )wc→c ' lnP(c ' , t)P (c ,t )

⟨ Stot ⟩ = ⟨ S sys⟩ + ⟨ Senv⟩

S=−kB∑c , c '

ξc , c ' ln[X c ' ]wc '→c

[X c ]wc→c '

S = −kB∑c , c '

ξc , c ' ln[X c ' ]

[X c]− kB∑

c , c '

ξc , c ' lnwc '→c

wc→c '

Brief summary of Schnakenbergs argument (4)Brief summary of Schnakenbergs argument (4)

⟨ Stot ⟩ = ⟨ S sys⟩ + ⟨ Senv⟩

⟨ Senv(t)⟩ = ∑c , c '∈Ωsys

P(c , t)wc→c ' lnwc→c '

wc '→c

S=−kB∑c , c '

ξc , c ' ln[X c ' ]wc '→c

[X c ]wc→c '

S = −kB∑c , c '

ξc , c ' ln[X c ' ]

[X c]− kB∑

c , c '

ξc , c ' lnwc '→c

wc→c '

⟨ Ssys(t)⟩ = − ∑c , c '∈Ωsys

P (c ,t )wc→c ' lnP(c ' , t)P (c ,t )

Environmental entropy productionEnvironmental entropy production

⟨ Senv(t) ⟩ = ∑c ,c '∈Ωsys

P(c , t )wc ,→c ' lnwc→c '

wc '→ c

Senv(t) = ∑j

δ(t−t j) lnwc j−1→ c j

wc j→c j−1

Important consequence:

Irreversible transitions do not exist.In Nature, there are no „absorbing states“.

totTotal state space

sysSystem state space

EnvironmentSystem

drive

Explaining entropy productionExplaining entropy productionin terms of microstatesin terms of microstates

Simplest example:Simplest example:

Stochastic clock in a stationary state

Counting the number of cycles, we may think of a linear chain of transitions

Each configuration corresponds to a certain number of configurations of the environment.

Assume equal rates among all transitions

Subsystem is driven by an entropic force.

wcc '

wc 'c

=N c ' N c

Environmental entropy production 

wcc '

wc 'c

=N env c '

N env c S env = −ln N env c '

ln N envc

S env = lnwcc '

wc 'c

Senv(t) = ∑j

δ(t−t j) lnωc j−1→ c j

ωc j→ c j−1

Question

Under which conditions is this formula correct?

  Answer:Answer:

●  The formula is correct if the environment The formula is correct if the environment       equilibrates instantaneouslyequilibrates instantaneously after each transition. after each transition.

●  In realistic systems this is not necessarily true.In realistic systems this is not necessarily true.

●  The formula could provide an upper bound in the   The formula could provide an upper bound in the     long­time limit (ongoing research)long­time limit (ongoing research)

Senv(t) = ∑j

δ(t−t j) lnωc j−1→ c j

ωc j→ c j−1

Question:Question: Under which conditions is Under which conditions is this formula correct? this formula correct?

// Example: biased random walkconst double p=0.3;int x=0; double S_env=0;...if (rnd()<p)

{x++;S_env += ln(p)/ln(1­p)}

else{x­­;S_env ­= ln(p)/ln(1­p);}

Environmental entropy production is easily accessible in numerical simulations.

Whenever the configuration changes, simply add  lnwc→c '

wc '→c

p1-p

Ssys(t ) = −P (c (t ), t)P (c (t ), t)

−∑j

δ(t−t j) lnP(c j , t)

P(c j−1 , t)

Senv(t) = ∑j

δ(t−t j) lnωc j→ c j+1

ωc j+1→ c j

Stot (t ) = −P (c (t ), t)P (c (t ), t)

−∑j

δ(t−t j) lnP (c j , t)wc j−1→c j

P (c j−1 , t)wc j→c j−1

No entropy production in the stationary state       Detailed balance↔

Two equivalent definitions of detailed balance:Two equivalent definitions of detailed balance:

Probability currents in the stationary state cancel 

pairwise:

∀ c ,c '∈ :

P cwcc ' = P c ' wc 'c

For each closed stochastic path                                    

the product of all rates along this path is equal to the product of 

the rates in reverse direction

wc1c2wc2c3

...wcN−1cNwcNc1

=

wcNcN−1wcN−1cN−2

...wc2c1wc1cN

c1 c2 ...cN c1

●   does not rely on P(c)●   difficult to prove●   easy to disprove

●   requires knowledge of P(c)●   easy to prove 

2. Fluctuation theorem revisited

t

Stot(t)

t

ΔStot(t)

Second law: 

but it fluctuates ­ sometimes even in opposite direction

⟨ S tot ⟩ ≥0

t

Stot(t)

t

ΔStot(t)

P(ΔStot)

ΔStot

t

Stot(t)

t

ΔStot(t)

P(ΔStot)

ΔStot

P S tot

P − S tot= e S tot

Fluctuation theorem:

To prove the fluctuation theorem, 

1)   prove it for a single transition c↔c'

2)   show that it will hold for any sequence       of transitions

YAP ­ yet another proofYAP ­ yet another proofof the fluctuation relationof the fluctuation relation

P (Δ Stot)

P(−ΔS tot)= eΔ S tot

c c'

First step:Consider a single transition c↔c'

P(ΔStot)

ΔStot

First step:Consider a single transition c↔c'

c c'

P(ΔStot)

ΔStot

S tot = lnP cwcc '

P c ' wc 'c

P S tot ∝ P c wcc '

P − S tot ∝ P c ' wc 'c

P S tot

P − S tot = exp S tot

c c'

Fluctuation theorem holds trivially !

Second step: Show that the FR holds for any sequence.

 Prove invariance under → convolution:

f x = f −x e x

g x =g −x e x

f∗g x = ∫ f ( y)g(x− y)dy

= ∫ f (− y)e y g(−x+ y)ex− y

= ex∫ f (−y)g( y−x)dy

= ex∫ f ( y)g(− y−x)dy = ex(f∗g)(−x)

Fluctuation relation

­ holds exactly for the total entropy

­ holds approximately for the environmental entropyproduction in a non­equilibrium steady state in

the long time limit

P(Δ Senv)

P(−Δ Senv)≈ exp(Δ Senv)

P S tot

P − S tot= exp S tot

Distribution itself is system­dependent

3. Entropy production and renormalization

Arrow can be interpreted as ' time'

Contact process:

A → 2A2A → AA → 0

Example: Directed percolation (DP)Example: Directed percolation (DP)

Bonds openwith probability p

Toy model for epidemic spreading

Absorbing states         Infinite entropy production↔

Renormalization scheme for DP by logical ORRenormalization scheme for DP by logical OR

Let               be the probability to find adjacent blocks of size mat time t in the bit pattern p.

Example:

P101(5)

000101001010000001011010110

1     1    0    1    1

Pp(m)(t )

Let               be the probability to find adjacent blocks of size mat time t in the bit pattern p.

Example:

In a critical DP process                increases with timewhile all other decrease with time.

                                   saturates as 

P101(5)

000101001010000001011010110

1     1    0    1    1

Pp(m)(t )

P000(m) (t )

S p(m)(t) :=

P p(m)(t)

1−P000(m)(t)

t→∞

Perform two limits:Perform two limits:

1. Take time  

2. Take block size  

Observation: These quantities are universal.

S p(m) := lim

t→∞S p(m)(t) =

Pp(m)(t)

1−P000(m)(t )

t→∞

m→∞

S p* := lim

m→∞S p(m)

t→∞

m→∞

Useful for:

● Verification whether a given model belongs to DP● Definition of a „clean“ contact process

Number of bits Number of univ. quantities

2 2

3 5

4 9

5 17

space

time

space

time

space

time

space

time

space

time

0 1 →

1 0 →

space

time

0 1 →

1 0 →

space

time

0 1 →

1 0 →

space

time

0 1 →

1 0 →

space

time

0 1 →

1 0 →

w100

w111

Effective transition rates  wp in the coarse­grained dynamics

There are

● Reversible transitions 110   111↔

● Irreversible transitions 010   000↔

● Impossible transitions 000   010→

The allowed transitions are expected to decreasewith increasing block size.

Example: Effective 3­bit ratesExample: Effective 3­bit rates

Example: Effective 3­bit ratesExample: Effective 3­bit rates

m=4

m=32

t→∞

Example: Effective 3­bit ratesExample: Effective 3­bit rates

Example: Effective 3­bit ratesExample: Effective 3­bit rates

Observation:

1.   The irreversible rates decrease faster than       the reversible rates with increasing block size.

w prev∼m−2 , w p

irr∼m−2.6

Example: Effective 3­bit Example: Effective 3­bit currentscurrents

Example: Effective 3­bit Example: Effective 3­bit currentscurrents

m→∞

Observation:Observation:

1.   The irreversible currents decrease faster than       the reversible currents with increasing block size.

2.  The reversible currents approach each other     as if they would satisfy detailed balance     in the limit  

J prev∼m−2 , J p

irr∼m−2.6

m→∞

Summary

●   The commonly accepted formula for environmental     entropy production holds only if the environment    equilibrates instantaneously.

●  The fluctuation theorem is a property that it is invariant   under convolution.

● It is very difficult (although not impossible) to find other    physical quantities which obey the fluctuation theorem.

● Directed percolation has infinite entropy production.  Under block renormalization, however, the currents of   irreversible transitions vanish faster while reversible transitions  seem to approach detailed balance.