Entropy production and fluctuation phenomena in nonequilibrium - - PowerPoint PPT Presentation

Entropy production and fluctuation phenomena in nonequilibrium systems

Haye Hinrichsen Faculty for Physics and Astronomy University of Würzburg, Germany Workshop on Large Fluctuations in Non-Equilibrium Systems MPIPKS Dresden, July 2011

In collaboration with:

Andre Barato, ICTP, Trieste, Italy Urna Basu, SAHA Institute, Kolkata, India Raphael Chetrite, Lyon and CNRS Christian Gogolin, Potsdam Peter Janotta, Würzburg David Mukamel, Weizmann Insititute, Israel

Non-equilibrium Dynamics, Thermalization and Entropy Production

• H. Hinrichsen, C. Gogolin, and P. Janotta
• J. Phys.: Conf. Ser. 297 012011 (2011)

Entropy production and fluctuation relations for a KPZ interface

• A. C. Barato, R. Chetrite, H. Hinrichsen, and D. Mukamel
• J. Stat. Mech.: Theor. Exp. P10008 (2010)

Outline 1) Introduction to entropy production 2) Fluctuation theorem revisited 3) Entropy production and renormalization

Nonequilibrium systems Nonequilibrium systems

T1 T2 μ1 μ2

Flow of heat … typically driven systems Flow of particles

Environment

Nonequilibrium systems Nonequilibrium systems

System

Environment

Nonequilibrium systems Nonequilibrium systems

System

drive entropy

Models of classical nonequilibrium systems Models of classical nonequilibrium systems

System

entropy Model

Models of classical nonequilibrium systems Models of classical nonequilibrium systems

System

entropy

Set of configurations Ωsys (state space)

configurations c∈sys Model

Models of classical nonequilibrium systems Models of classical nonequilibrium systems

System

entropy Model Irreversible dynamics by spontaneous transitions at rate

cc'

wcc'

Ωsys

Configurational entropy Environmental entropy Total entropy

S syst = −ln Pc ,t S envt Stot(t)=Ssys(t)+Senv(t)

Environment

System

drive entropy

Actual time evolution: Sequence of transitions (stochastic path) at times Our partial knowledge: Probability distribution P(c,t) evolving deterministically by the master equation.

c1c2c3...c N t1 ,t 2 ,t3 ,... ,t N

〈Ssys(t)〉 = −∑c∈Ωsys P(c,t)ln P(c,t)

Ssys(t) = −ln P(c(t),t) d dt P(c,t) = ∑

c '∈Ω

P(c' ,t)wc' c−P(c,t)wcc'

Configurational entropy Mean entropy

Entropy of the system Entropy 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 system Entropy of the system

Change of conf. entropy

˙ Ssys(t) = − ˙ P(c(t),t) P(c(t),t)−∑

j

δ(t−t j)ln P(c j,t) P(c j−1,t)

〈 ˙

Ssys(t)〉 = − ∑

c,c'∈Ωsys

P(c,t)wc →c' ln P(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 Stot(t)=Ssys(t)+Senv(t)

Environment

System

drive entropy

? ?

˙ Senv(t) = ∑

j

δ(t−t j)ln ωc j−1 →c j ωc j →c j−1

Andrieux and Gaspard, J. Chem. Phys. 2004

• U. Seifert, PRL 2005

Commonly accepted formula for the Commonly accepted formula for the environmental entropy environmental 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 system Fictitious chemical system d dt P(c ,t) = ∑

c'∈Ω

(P(c' ,t)wc' →c−P(c ,t)wc →c')

Isothermal / isochroric → minimize F.

Extent of reaction Extent 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 flux Thermodynamic flux Conjugate thermodynamic force Conjugate thermodynamic force Extent of reaction Extent of reaction ξ ξcc′

cc′

Chemical a nity ffi Chemical a nity ffi

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'

Acc' ˙ N cc'

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

In the stationary state we have With . Hence turns into

˙ F=∑

cc '

Acc ' ˙ ξcc '

E,T constant

Brief summary of Schnakenbergs argument (4) Brief summary of Schnakenbergs argument (4) ˙ S=−k B∑

c,c'

ξc ,c' ln [Xc ']wc' →c [Xc]wc →c ' ˙ S = −k B∑

c,c'

ξc ,c' ln [ X c'] [ X c] − k B∑

c,c'

ξc ,c' ln wc' →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' ln P(c' ,t) P(c ,t)

〈 ˙ Stot〉 = 〈 ˙ Ssys〉 + 〈 ˙ Senv〉 ˙ S=−k B∑

c,c'

ξc ,c' ln [Xc ']wc' →c [Xc]wc →c ' ˙ S = −k B∑

c,c'

ξc ,c' ln [ X c'] [ X c] − k B∑

c,c'

ξc ,c' ln wc' →c wc →c'

Brief summary of Schnakenbergs argument (4) Brief summary of Schnakenbergs argument (4) 〈 ˙ Stot〉 = 〈 ˙ Ssys〉 + 〈 ˙ Senv〉

〈 ˙

Senv(t)〉 =

∑

c,c'∈Ωsys

P(c,t)wc →c 'ln wc →c ' wc ' →c

˙ S=−k B∑

c,c'

ξc ,c' ln [Xc ']wc' →c [Xc]wc →c ' ˙ S = −k B∑

c,c'

ξc ,c' ln [ X c'] [ X c] − k B∑

c,c'

ξc ,c' ln wc' →c wc →c'

〈 ˙

Ssys(t)〉 = − ∑

c,c'∈Ωsys

P(c,t)wc →c' ln P(c' ,t) P(c ,t)

Environmental entropy production Environmental entropy production

〈 ˙

Senv(t)〉 =

∑

c,c'∈Ωsys

P(c,t)wc, →c' ln wc →c' wc' →c ˙ Senv(t) = ∑

j

δ(t−t j)ln wc j−1 → c j wc j→c j−1

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

tot

Total state space

sys

System state space

Environment

System

drive

Explaining entropy production Explaining entropy production in terms of microstates in 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.

wcc' wc' c = N c'  N c

Environmental entropy production

wcc' wc' c = N envc'  N envc  S env = −ln N envc' ln N envc  S env = ln wcc' wc' c

˙ Senv(t) = ∑

j

δ(t−t j)ln ωc 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 instantaneously equilibrates 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 long-time limit (ongoing research) long-time limit (ongoing research)

˙ Senv(t) = ∑

j

δ(t−t j)ln ωc 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 walk const double p=0.3; int x=0; double S_env=0; ... if (rnd()<p) { x++; S_env += ln(p)/ln(1-p) } else { x--; S_env -= ln(p)/ln(1-p); }

Environmental entropy production is easily accessible in numerical simulations.

Whenever the configuration changes, simply add

ln wc→c' wc' →c

p 1-p

˙ Ssys(t) = − ˙ P(c(t),t) P(c(t),t)−∑

j

δ(t−t j)ln P(c j,t) P(c j−1,t)

˙ Senv(t) = ∑

j

δ(t−t j)ln ωc j →c j+1 ωc j+1→c j

˙ Stot(t) = − ˙ P(c(t),t) P(c(t),t)−∑

j

δ(t−t j)ln P(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cwcc' = 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

wc1c2wc2c3...wcN −1cN wc N c1 = wcN cN −1wcN −1cN −2...wc2c1wc1cN

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

 Stot

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

• f transitions

YAP - yet another proof YAP - yet another proof

• f the fluctuation relation
• f the fluctuation relation

P(Δ Stot) P(−Δ Stot) = e

Δ Stot

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 = ln Pcwcc' Pc' wc' c P S tot ∝ Pcwcc' 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)e x− y

= e

x∫ f (−y)g( y−x)dy

= e

x∫ f ( y)g(−y−x)dy = e x (f ∗g)(−x)

Fluctuation relation

• holds exactly for the total entropy
• holds approximately for the environmental entropy

production in a non-equilibrium 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 system-dependent

• 3. Entropy production and renormalization

Arrow can be interpreted as ' time'

Contact process: A → 2A 2A → A A → 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 OR Renormalization scheme for DP by logical OR

Let be the probability to find adjacent blocks of size m at 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 m at time t in the bit pattern p. Example: In a critical DP process increases with time while 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 coarse-grained dynamics

There are

• Reversible transitions

110 111 ↔

• Irreversible transitions

010 000 ↔

• Impossible transitions

000 010 → The allowed transitions are expected to decrease with increasing block size. Example: Effective 3-bit rates Example: Effective 3-bit rates

Example: Effective 3-bit rates Example: Effective 3-bit rates

m=4 m=32

t →∞

Example: Effective 3-bit rates Example: Effective 3-bit rates

Example: Effective 3-bit rates Example: Effective 3-bit rates Observation:

• 1. The irreversible rates decrease faster than

the reversible rates with increasing block size.

w p

rev∼m −2,

w p

irr∼m −2.6

Example: Effective 3-bit Example: Effective 3-bit currents currents

Example: Effective 3-bit Example: Effective 3-bit currents currents

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 p

rev∼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.