The Min/Max-Entropy Production Principles Introduction Markov - PowerPoint PPT Presentation

The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný The Min/Max-Entropy Production Principles Introduction Markov Chains from a Dynamical Fluctuation Law Electrical Networks Landauer’s Counterex. C. Maes 1 cný 2 Dynamical Fluctuations K. Netoˇ General considerations Donsker-Varadhan Theory Equilibrium Dynamics 1 Instituut voor Theoretische Fysica Close-to-Equilibrium K. U. Leuven Macroscopic Limit Introduction 2 Institute of Physics Free Particles Model Academy of Sciences of the Czech Republic El. Circuits Revisited MinEP principle Conclusions Mathematical Physics Days XIII, Leuven 28-29 September 2006

Introduction The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains ◮ Thermal equilibrium distribution is characterized by the Electrical Networks Landauer’s Counterex. Gibbs variational principle Dynamical Fluctuations ◮ proves useful in analyzing infinite-volume systems, General considerations Donsker-Varadhan Theory phase transitions,... Equilibrium Dynamics Close-to-Equilibrium ◮ No general variational characterization of Macroscopic Limit nonequilibrium stationary states is known! Introduction Free Particles Model ◮ Yet, there are various approximative principles: El. Circuits Revisited MinEP principle ◮ Minimum & maximum entropy production principles Conclusions ◮ Either restricted to close-to-equilibrium regime or uncontrollable ◮ Even close to equilibrium counterexamples are known! ◮ What is the status of these principles? ◮ Can we understand more or even go beyond?

Introduction The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains ◮ Thermal equilibrium distribution is characterized by the Electrical Networks Landauer’s Counterex. Gibbs variational principle Dynamical Fluctuations ◮ proves useful in analyzing infinite-volume systems, General considerations Donsker-Varadhan Theory phase transitions,... Equilibrium Dynamics Close-to-Equilibrium ◮ No general variational characterization of Macroscopic Limit nonequilibrium stationary states is known! Introduction Free Particles Model ◮ Yet, there are various approximative principles: El. Circuits Revisited MinEP principle ◮ Minimum & maximum entropy production principles Conclusions ◮ Either restricted to close-to-equilibrium regime or uncontrollable ◮ Even close to equilibrium counterexamples are known! ◮ What is the status of these principles? ◮ Can we understand more or even go beyond?

Introduction The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains ◮ Thermal equilibrium distribution is characterized by the Electrical Networks Landauer’s Counterex. Gibbs variational principle Dynamical Fluctuations ◮ proves useful in analyzing infinite-volume systems, General considerations Donsker-Varadhan Theory phase transitions,... Equilibrium Dynamics Close-to-Equilibrium ◮ No general variational characterization of Macroscopic Limit nonequilibrium stationary states is known! Introduction Free Particles Model ◮ Yet, there are various approximative principles: El. Circuits Revisited MinEP principle ◮ Minimum & maximum entropy production principles Conclusions ◮ Either restricted to close-to-equilibrium regime or uncontrollable ◮ Even close to equilibrium counterexamples are known! ◮ What is the status of these principles? ◮ Can we understand more or even go beyond?

Introduction The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains ◮ Thermal equilibrium distribution is characterized by the Electrical Networks Landauer’s Counterex. Gibbs variational principle Dynamical Fluctuations ◮ proves useful in analyzing infinite-volume systems, General considerations Donsker-Varadhan Theory phase transitions,... Equilibrium Dynamics Close-to-Equilibrium ◮ No general variational characterization of Macroscopic Limit nonequilibrium stationary states is known! Introduction Free Particles Model ◮ Yet, there are various approximative principles: El. Circuits Revisited MinEP principle ◮ Minimum & maximum entropy production principles Conclusions ◮ Either restricted to close-to-equilibrium regime or uncontrollable ◮ Even close to equilibrium counterexamples are known! ◮ What is the status of these principles? ◮ Can we understand more or even go beyond?

Introduction The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains ◮ Thermal equilibrium distribution is characterized by the Electrical Networks Landauer’s Counterex. Gibbs variational principle Dynamical Fluctuations ◮ proves useful in analyzing infinite-volume systems, General considerations Donsker-Varadhan Theory phase transitions,... Equilibrium Dynamics Close-to-Equilibrium ◮ No general variational characterization of Macroscopic Limit nonequilibrium stationary states is known! Introduction Free Particles Model ◮ Yet, there are various approximative principles: El. Circuits Revisited MinEP principle ◮ Minimum & maximum entropy production principles Conclusions ◮ Either restricted to close-to-equilibrium regime or uncontrollable ◮ Even close to equilibrium counterexamples are known! ◮ What is the status of these principles? ◮ Can we understand more or even go beyond?

Introduction The Min-MaxEP Principles from a Dynamical Fluctuation Law Maes, Netoˇ cný Introduction Markov Chains ◮ Thermal equilibrium distribution is characterized by the Electrical Networks Landauer’s Counterex. Gibbs variational principle Dynamical Fluctuations ◮ proves useful in analyzing infinite-volume systems, General considerations Donsker-Varadhan Theory phase transitions,... Equilibrium Dynamics Close-to-Equilibrium ◮ No general variational characterization of Macroscopic Limit nonequilibrium stationary states is known! Introduction Free Particles Model ◮ Yet, there are various approximative principles: El. Circuits Revisited MinEP principle ◮ Minimum & maximum entropy production principles Conclusions ◮ Either restricted to close-to-equilibrium regime or uncontrollable ◮ Even close to equilibrium counterexamples are known! ◮ What is the status of these principles? ◮ Can we understand more or even go beyond?

Example 1: Markov chains The Min-MaxEP Principles from a Dynamical Fluctuation Notation Law Consider a finite-state continuous-time Markov process η t : Maes, Netoˇ cný ◮ Rates λ ( x , y ) Introduction Markov Chains ◮ Ergodicity assumption = ⇒ unique stationary Electrical Networks Landauer’s Counterex. distribution ρ : Dynamical Fluctuations General considerations � Donsker-Varadhan Theory [ ρ ( x ) λ ( x , y ) − ρ ( y ) λ ( y , x )] = 0 Equilibrium Dynamics Close-to-Equilibrium x Macroscopic Limit Introduction Free Particles Model ◮ Alternatively, the process given by the generator: El. Circuits Revisited MinEP principle � Conclusions λ ( x , y )[ f ( η xy ) − f ( η )] ( Lf )( η ) = y � = x so that E µ [ f ( η t )] = µ ( e tL f ) = µ t ( f ) ◮ Describes a general nonequilibrium dynamics of a thermodynamically open system

Example 1: Markov chains The Min-MaxEP Principles from a Dynamical Fluctuation Notation Law Consider a finite-state continuous-time Markov process η t : Maes, Netoˇ cný ◮ Rates λ ( x , y ) Introduction Markov Chains ◮ Ergodicity assumption = ⇒ unique stationary Electrical Networks Landauer’s Counterex. distribution ρ : Dynamical Fluctuations General considerations � Donsker-Varadhan Theory [ ρ ( x ) λ ( x , y ) − ρ ( y ) λ ( y , x )] = 0 Equilibrium Dynamics Close-to-Equilibrium x Macroscopic Limit Introduction Free Particles Model ◮ Alternatively, the process given by the generator: El. Circuits Revisited MinEP principle � Conclusions λ ( x , y )[ f ( η xy ) − f ( η )] ( Lf )( η ) = y � = x so that E µ [ f ( η t )] = µ ( e tL f ) = µ t ( f ) ◮ Describes a general nonequilibrium dynamics of a thermodynamically open system

Example 1: Markov chains The Min-MaxEP Principles from a Dynamical Fluctuation Equilibrium dynamics Law Maes, Netoˇ cný Detailed balance assumption: ρ ( x ) λ ( x , y ) = ρ ( y ) λ ( y , x ) Introduction Markov Chains ◮ Describes an equilibrium dynamics Electrical Networks Landauer’s Counterex. ◮ closed system for ρ the counting measure Dynamical Fluctuations ◮ Relaxing to an equilibrium with a bath for ρ a Gibbs General considerations Donsker-Varadhan Theory measure Equilibrium Dynamics Close-to-Equilibrium Defining the entropy Macroscopic Limit Introduction µ ( x ) log µ ( x ) Free Particles Model � S ( µ ) = − S ( µ | ρ ) = − El. Circuits Revisited ρ ( x ) MinEP principle x Conclusions the entropy production rate is E ( µ ) = d S ( µ t ) � µ ( x ) λ ( x , y ) log µ ( x ) ρ ( y ) � � t = 0 = � d t µ ( y ) ρ ( x ) x , y � = x ◮ By convexity, σ ( µ ) ≥ 0 and the equality is only for µ = ρ