Nonequilibrium variational principles Nonequilibrium variational - PowerPoint PPT Presentation

Nonequilibrium variational principles Nonequilibrium variational principles from dynamical fluctuations from dynamical fluctuations Karel Neto č ný Institute of Physics AS CR MRC, Warwick University, 18 May 2010

To be discussed Min- and Max-entropy production principles: various examples From variational principles to fluctuation laws: equilibrium case Static versus dynamical fluctuations Onsager-Machlup equilibrium dynamical fluctuation theory Stochastic models of nonequilibrium Conclusions, open problems, outlook,... In collaboration with C. Maes, B. Wynants, and S. Bruers (K.U.Leuven, Belgium)

Motivation : Modeling Earth climate [Ozawa et al, Rev. Geoph. 41 (2003) 1018]

Linear electrical networks explaining MinEP/MaxEP principles Kirchhoff’s loop law: � � U jk = E jk k k Entropy production rate: � U � jk σ ( U ) = βQ ( U ) = β R jk j,k MinEP principle: U �� Stationary values of voltages minimize the entropy production rate Not valid under inhomogeneous temperature!

Linear electrical networks explaining MinEP/MaxEP principles Kirchhoff’s current law: � J jk = 0 j Entropy production rate: � R jk J � σ ( J ) = βQ ( J ) = β jk j,k Work done by sources: U �� � W ( J ) = E jk J jk jk (Constrained) MaxEP principle: Stationary values of currents maximize the entropy production under constraint Q ( J ) = W ( J )

Linear electrical networks summary of MinEP/MaxEP principles MaxEP principle I + Current law Current law U + Loop law Loop law + U, I MinEP principle Generalized variational principle

From principles to fluctuation laws Questions and ideas How to go beyond approximate and ad hoc thermodynamic principles? Inspiration from thermostatics: Equilibrium variational principles are intimately related to the structure of equilibrium fluctuations Is there a nonequilibrium analogy of thermodynamical fluctuation theory?

From principles to fluctuation laws Equilibrium fluctuations H ( x ) = Ne H ( x ) = Ne add field P ( M ( x ) = Nm ) = e N � s � e,m � − s eq � e �� M ( x ) = Nm eq ( e ) Probability of fluctuation Typical value H h ( x ) = H ( x ) − hM ( x ) = N [ e − h m ] The fluctuation made typical! s ( e, m ) = s h eq ( e − hm )

From principles to fluctuation laws Equilibrium fluctuations Variational functional Fluctuation functional Thermodynamic potential Entropy (Generalized) free energy

From principles to fluctuation laws Static versus dynamical fluctuations Empirical time average: � T m T = � � � m ( x t ) dt T Ergodic property: m T → m eq ( e ) , � T → ∞ H ( x ) = Ne Dynamical fluctuations: m T = m ) = e − T I � m � P ( � Static: τ → ∞ I � ∞ � ( m ) = s ( e ) − s ( e,m ) Interpolating between static and dynamical fluctuations: � � � � n = e − n I � τ � � m � Dynamic: τ → 0 P k �� m ( x τk ) = m n

Effective model of macrofluctuations Onsager-Machlup theory � � R Dynamics: R dm t = − s m t dt + N dB t � Ns m � P ( m ∞ = m ) ∝ e − � Equilibrium: S ( m ) − S (0) Path distribution: � � T � dm t � � � − N R dt + s P ( ω ) = exp R m t � � �

Effective model of macrofluctuations Onsager-Machlup theory � � R Dynamics: R dm t = − sm t dt + N dB t Path distribution: � � T � dm t � � � − N R dt + s P ( ω ) = exp R m t � � � Dynamical fluctuations: � 8 R m � � − T Ns � P ( � m T = m ) = P ( m t = m ; 0 ≤ t ≤ T ) = exp (Typical immediate) entropy production rate: = Ns � σ ( m ) = dS � m t � � R m � dt

Effective model of macrofluctuations Onsager-Machlup theory � � R Dynamics: R dm t = − sm t dt + N dB t Path distribution: � � T � dm t � � � − N R dt + s P ( ω ) = exp R m t � � � Dynamical fluctuations: � 8 R m � � − T Ns � P ( � m T = m ) = P ( m t = m ; 0 ≤ t ≤ T ) = exp I ( m ) = � � σ ( m ) (Typical immediate) entropy production rate: = Ns � σ ( m ) = dS � m t � � R m � dt

Towards general theory Equilibrium Nonequilibrium Closed Open Hamiltonian dynamics Stochastic dynamics Microscopic Macroscopic

Linear electrical networks revisited Dynamical fluctuations Fluctuating dynamics: E = U + R � J + E f E � U + U − E f J = C ˙ U � R � E f E f R � R � Johnson-Nyquist noise: � � � C E f � R t = β ξ t white noise Empirical time average: � T � U T = � � U t dt T Dynamical fluctuation law: � � R � + � E − U � � U � E � T log P ( � β � β � � R � � R � � − � U T = U ) = � − � β � R � � β � R � R � R � � R �

Linear electrical networks revisited Dynamical fluctuations Fluctuating dynamics: E = U + R � J + E f E � U + U − E f J = C ˙ U � R � E f E f R � R � Johnson-Nyquist noise: � � � C E f � R t = β ξ t white noise Empirical time average: total dissipated � T � U T = � � U t dt heat T Dynamical fluctuation law: � � R � + � E − U � � U � E � T log P ( � β � β � � R � � R � � − � U T = U ) = � − � β � R � � β � R � R � R � � R �

Stochastic models of nonequilibrium breaking detailed balance Local detailed balance: k ( x log k � x,y � , y ) k � y,x � = ∆ s ( x, y ) = − ∆ s ( y, x ) x y k ( y, x ) Global detailed balance generally broken: ∆ s ( x, y ) = s ( y ) − s ( x ) + ǫF ( x, y ) Markov dynamics: � � � dρ t ( x ) = ρ t ( y ) k ( y, x ) − ρ t ( x ) k ( x, y ) dt y

Stochastic models of nonequilibrium breaking detailed balance Local detailed balance: k ( x log k � x,y � , y ) k � y,x � = ∆ s ( x, y ) = − ∆ s ( y, x ) x y entropy change k ( y, x ) in the environment Global detailed balance generally broken: ∆ s ( x, y ) = s ( y ) − s ( x ) + ǫF ( x, y ) Markov dynamics: � � � dρ t ( x ) = ρ t ( y ) k ( y, x ) − ρ t ( x ) k ( x, y ) dt y

Stochastic models of nonequilibrium breaking detailed balance Local detailed balance: k ( x log k � x,y � , y ) k � y,x � = ∆ s ( x, y ) = − ∆ s ( y, x ) x y entropy change k ( y, x ) in the environment Global detailed balance generally broken: ∆ s ( x, y ) = s ( y ) − s ( x ) + ǫF ( x, y ) breaking term Markov dynamics: � � � dρ t ( x ) = ρ t ( y ) k ( y, x ) − ρ t ( x ) k ( x, y ) dt y

Stochastic models of nonequilibrium entropy production Entropy of the system: � S ( ρ ) = − ρ ( x ) log ρ ( x ) k ( x , y ) x x y Mean currents: k ( y, x ) J ρ ( x, y ) = ρ ( x ) k ( x, y ) − ρ ( y ) k ( y, x ) � �� � ���� �� �������� ������� Mean entropy production rate: � σ ( ρ ) = dS ( ρ t ) + 1 J ρ ( x, y )∆ s ( x, y ) dt 2 � x,y � � ρ ( x ) k ( x, y ) log ρ ( x ) k ( x, y ) = ρ ( y ) k ( y, x ) x,y

Stochastic models of nonequilibrium entropy production Entropy of the system: S ( ρ ) = − � x ρ ( x ) log ρ ( x ) k ( x , y ) x y Entropy fluxes: k ( y, x ) J ρ ( x, y ) = ρ ( x ) k ( x, y ) − ρ ( y ) k ( y, x ) � �� � ���� �� �������� ������� Mean entropy production rate: � σ ( ρ ) = dS ( ρ t ) + 1 J ρ ( x, y )∆ s ( x, y ) dt 2 Warning : � x,y � Only for time-reversal � ρ ( x ) k ( x, y ) log ρ ( x ) k ( x, y ) symmetric observables! = ρ ( y ) k ( y, x ) ≥ 0 x,y

Stochastic models of nonequilibrium MinEP principle (“Microscopic”) MinEP principle: k ( x , y ) In the first order approximation around x y detailed balance k ( y, x ) σ ( ρ ) = min ⇒ ρ = ρ s + O ( ǫ � ) Can we again recognize entropy production as a fluctuation functional?

Stochastic models of nonequilibrium dynamical fluctuations Empirical occupation times: � T p T ( x ) = � � � χ ( ω t = x ) dt k ( x , y ) T x y Ergodic theorem: k ( y, x ) p T ( x ) → ρ s ( x ) , � T → ∞ Fluctuation law for occupation times? p T = p ) = e − T I � p � P (� Note : I ( ρ s ) = 0

Stochastic models of nonequilibrium dynamical fluctuations Idea : Make the empirical distribution typical by modifying dynamics: → k v ( x, y ) = k ( x, y ) e � v � y � − v � x �� / � k ( x, y ) − The “field” v is such that distribution p is stationary distribution for the modified dynamics: � � � p ( y ) k v ( y, x ) − p ( x ) k v ( x, y ) = 0 y Comparing both processes yields the fluctuation law: � � � I ( p ) = p ( x ) k ( x, y ) − k v ( x, y ) x,y

Recall Equilibrium fluctuations H ( x ) = Ne H ( x ) = Ne add field P ( M ( x ) = Nm ) = e N � s � e,m � − s eq � e �� M ( x ) = Nm eq ( e ) Probability of fluctuation Typical value H h ( x ) = H ( x ) − hM ( x ) = N [ e − h m ] The fluctuation made typical! s ( e, m ) = s h eq ( e − hm )