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

I n 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 T 1 T 2 Flow of heat μ 1 μ 2 Flow of particles … typically driven systems

Nonequilibrium systems Nonequilibrium systems System Environment

Nonequilibrium systems Nonequilibrium systems drive System Environment 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 Ω sys (state space) Model configurations c ∈ sys

Models of classical nonequilibrium systems Models of classical nonequilibrium systems System entropy Ω sys Model Irreversible dynamics by spontaneous transitions w c  c' c  c' at rate

drive Environment System entropy S sys  t  = − ln P  c ,t  Configurational entropy S env  t  Environmental entropy S tot ( t )= S sys ( t )+ S env ( t ) Total entropy

Entropy of the system Entropy of the system Actual time evolution: Our partial knowledge: Sequence of transitions Probability distribution P(c,t) (stochastic path) evolving deterministically by the master equation. c 1  c 2  c 3  ...  c N d dt P ( c,t ) = ∑ t 1 ,t 2 ,t 3 , ... ,t N P ( c' ,t ) w c' c − P ( c,t ) w cc' at times c ' ∈Ω S sys ( t ) = − ln P ( c ( t ) ,t ) Configurational entropy 〈 S sys ( t ) 〉 = − ∑ c ∈Ω sys P ( c,t ) ln P ( c,t ) Mean entropy

Entropy of the system Entropy of the system S sys ( t ) = − ln P ( c ( t ) ,t ) Configurational entropy S sys ( t ) = − ˙ δ( t − t j ) ln P ( c j ,t ) P ( c ( t ) ,t ) P ( c ( t ) ,t )− ∑ ˙ Change of conf. entropy P ( c j − 1 ,t ) j 〈 S sys ( t ) 〉 = − ∑ P ( c,t ) ln P ( c,t ) Mean entropy c ∈Ω sys P ( c,t ) w c → c' ln P ( c ,t ) S sys ( t ) 〉 = − ∑ 〈 ˙ Change of mean entropy P ( c' ,t ) c,c' ∈Ω sys

drive Environment System entropy S sys  t  = − ln P  c ,t  Configurational entropy ? ? S env  t  Environmental entropy S tot ( t )= S sys ( t )+ S env ( t ) Total entropy

Commonly accepted formula for the Commonly accepted formula for the environmental entropy environmental entropy δ( t − t j ) ln ω c j − 1 → c j S env ( t ) = ∑ ˙ ω c j → c j − 1 j Where does it come from? Andrieux and Gaspard, J. Chem. Phys. 2004 U. Seifert, PRL 2005

1976

X 1 X 2 X N

P ( c ,t ) probability [ X i ] ( t ) concentration

Fictitious chemical system Fictitious chemical system Schnakenberg : The master equation d dt P ( c ,t ) = ∑ ( P ( c' ,t ) w c' → c − P ( c ,t ) w c → c' ) c' ∈Ω is mapped to a fictitious chemical system evolving according to the law of mass action (= mean field equation) Isothermal / isochroric → minimize F .

Brief summary of Schnakenbergs argument (1) Brief summary of Schnakenbergs argument (1) Extent of reaction = average number of Extent of reaction forward reactions c→c' minus backward reactions c'→c. Thermodynamic flux Conjugate thermodynamic force Conjugate thermodynamic force Thermodynamic flux ffi Extent of reaction Extent of reaction ξ ξ cc′ Chemical a Chemical a ffi nity cc′ nity

Brief summary of Schnakenbergs argument (2) Brief summary of Schnakenbergs argument (2) Compare F = ∑ ξ cc' = − ∑ A cc' ˙ ˙ ˙ A cc' N cc' with cc' cc' → Chemical affinity is chemical potential difference With and we arrive at:

Brief summary of Schnakenbergs argument (3) Brief summary of Schnakenbergs argument (3) In the stationary state we have F = ∑ A cc ' ˙ ˙ ξ cc ' With . Hence turns into cc ' E,T constant

Brief summary of Schnakenbergs argument (4) Brief summary of Schnakenbergs argument (4) ξ c ,c' ln [ X c ' ] w c' → c S =− k B ∑ ˙ [ X c ] w c → c ' c,c' ξ c ,c' ln [ X c' ] ξ c ,c' ln w c' → c − k B ∑ k B ∑ ˙ S = − [ X c ] w c → c' c,c' c,c'

Brief summary of Schnakenbergs argument (4) Brief summary of Schnakenbergs argument (4) ξ c ,c' ln [ X c ' ] w c' → c S =− k B ∑ ˙ [ X c ] w c → c ' c,c' ξ c ,c' ln [ X c' ] ξ c ,c' ln w c' → c − k B ∑ k B ∑ ˙ S = − [ X c ] w c → c' c,c' c,c' 〈 ˙ 〈 ˙ 〈 ˙ S tot 〉 = S sys 〉 + S env 〉 P ( c,t ) w c → c' ln P ( c' ,t ) S sys ( t ) 〉 = − ∑ 〈 ˙ P ( c ,t ) c,c' ∈Ω sys

Brief summary of Schnakenbergs argument (4) Brief summary of Schnakenbergs argument (4) ξ c ,c' ln [ X c ' ] w c' → c S =− k B ∑ ˙ [ X c ] w c → c ' c,c' ξ c ,c' ln [ X c' ] ξ c ,c' ln w c' → c − k B ∑ k B ∑ ˙ S = − [ X c ] w c → c' c,c' c,c' 〈 ˙ 〈 ˙ 〈 ˙ S tot 〉 = S sys 〉 + S env 〉 P ( c,t ) w c → c' ln P ( c' ,t ) S sys ( t ) 〉 = − ∑ 〈 ˙ P ( c ,t ) c,c' ∈Ω sys P ( c,t ) w c → c ' ln w c → c ' ∑ 〈 ˙ S env ( t ) 〉 = w c ' → c c,c' ∈Ω sys

Environmental entropy production Environmental entropy production P ( c,t ) w c, → c' ln w c → c' ∑ 〈 ˙ S env ( t ) 〉 = w c' → c c,c' ∈Ω sys w c j − 1 → c j S env ( t ) = ∑ ˙ δ( t − t j ) ln w c j → c j − 1 j Important consequence: Irreversible transitions do not exist. In Nature, there are no „absorbing states“.

 tot Total state space Explaining entropy production Explaining entropy production in terms of microstates in terms of microstates drive Environment System  sys System state space

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 . w c  c' = N  c'  N  c  w c'  c

Environmental entropy production = N env  c'  w c  c'  S env = − ln N env  c'  N env  c   ln N env  c  w c'  c  S env = ln w c  c' w c'  c

Question ω c j − 1 → c j S env ( t ) = ∑ ˙ δ( t − t j ) ln ω c j → c j − 1 j Under which conditions is this formula correct?

Question: Question: ω c j − 1 → c j S env ( t ) = ∑ ˙ Under which conditions is Under which conditions is δ( t − t j ) ln ω c j → c j − 1 this formula correct? this formula correct? j 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)

Environmental entropy production is easily accessible in numerical simulations. ln w c → c' Whenever the configuration changes, simply add w c' → c // Example: biased random walk const double p=0.3; int x=0; double S_env=0; ... 1-p p if (rnd()<p) { x++; S_env += ln(p)/ln(1-p) } else { x--; S_env -= ln(p)/ln(1-p); }

ω c j → c j + 1 S env ( t ) = ∑ ˙ δ( t − t j ) ln ω c j + 1 → c j j S sys ( t ) = − ˙ δ( t − t j ) ln P ( c j ,t ) P ( c ( t ) ,t ) P ( c ( t ) ,t )− ∑ ˙ P ( c j − 1 ,t ) j δ( t − t j ) ln P ( c j ,t ) w c j − 1 → c j S tot ( t ) = − ˙ P ( c ( t ) ,t ) P ( c ( t ) ,t )− ∑ ˙ P ( c j − 1 ,t ) w c j → c j − 1 j ↔ No entropy production in the stationary state Detailed balance

Two equivalent definitions of detailed balance: Two equivalent definitions of detailed balance: For each closed stochastic path Probability currents in the c 1  c 2  ...  c N  c 1 stationary state cancel pairwise: the product of all rates along this path is equal to the product of ∀ c ,c' ∈ : the rates in reverse direction P  c  w c  c' = P  c'  w c'  c w c 1  c 2 w c 2  c 3 ... w c N − 1  c N w c N  c 1 = w c N  c N − 1 w c N − 1  c N − 2 ... w c 2  c 1 w c 1  c N ● does not rely on P(c) ● requires knowledge of P(c) ● easy to prove ● difficult to prove ● easy to disprove

2. Fluctuation theorem revisited