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)

http://www.physik.uni-wuerzburg.de/%7Ehinrichsen/publications/source/100.pdf

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


System

Environment


System

drive

entropy

Models of classical nonequilibrium systemsModels of classical nonequilibrium systems

Systementropy

Model


Systementropy

Set of configurations Ωsys

(state space)

configurations c∈ sys

Model


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 '


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


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:


F = ∑cc '

Acc '˙ξcc ' = −∑

cc '

A cc '˙N cc '


In the stationary state we have

With . Hence turns intoF=∑cc '

Acc '˙ξcc '

E,T constant


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 )

⟨ Stot ⟩ = ⟨ S sys⟩ + ⟨ Senv⟩

S=−kB∑c , c '


[X c ]wc→c '

S = −kB∑c , c '

ξc , c ' ln[X c ' ]

[X c]− kB∑

c , c '


wc→c '


⟨ Stot ⟩ = ⟨ S sys⟩ + ⟨ Senv⟩

⟨ Senv(t)⟩ = ∑c , c '∈Ωsys

P(c , t)wc→c ' lnwc→c '

wc '→c

S=−kB∑c , c '


[X c ]wc→c '

S = −kB∑c , c '

ξc , c ' ln[X c ' ]

[X c]− kB∑

c , 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


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


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


m=4

m=32

t→∞



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.

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Entropy production and fluctuation phenomena in nonequilibrium