The Minimum Entropy Production Equilibrium Fluctuations Principle - PowerPoint PPT Presentation

The MinEP principle from a Dynamical Fluctuation Law Maes, Netoˇ cný The Minimum Entropy Production Equilibrium Fluctuations Principle from a Dynamical Closed System Open System Fluctuation Law Microscopic Level Conclusion Dynamical Fluctuations Donsker-Varadhan Theory Equilibrium Dynamics C. Maes 1 cný 2 Close-to-Equilibrium K. Netoˇ MinEP principle Main Result Onsager-Machlup Process 1 Instituut voor Theoretische Fysica Conclusions K. U. Leuven 2 Institute of Physics Academy of Sciences of the Czech Republic MEP in Physics and Biology, Split 6-7th July 2006

The MinEP principle Thesis and Contents from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations Closed System Open System The analysis of stationary fluctuations provides Microscopic Level a natural framework for various variational principles Conclusion Dynamical Fluctuations of statistical mechanics. Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process Conclusions

The MinEP principle Thesis and Contents from a Dynamical Fluctuation Law Maes, Netoˇ cný Equilibrium Fluctuations Closed System Open System The analysis of stationary fluctuations provides Microscopic Level a natural framework for various variational principles Conclusion Dynamical Fluctuations of statistical mechanics. Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process ◮ Introduction: Equilibrium fluctuations Conclusions ◮ Closed versus open systems ◮ Macro- versus micro-description ◮ Dynamical fluctuations ◮ Law of empirical occupation times ◮ The entropy production as the fluctuation rate ◮ Equilibrium versus non-equilibrium Markov dynamics ◮ Onsager-Machlup process

The MinEP principle Thermodynamic Equilibrium from a Dynamical Fluctuation Law what did we learn in basic courses? Maes, Netoˇ cný After long enough time any closed system is found in one Equilibrium Fluctuations of typical microstates. Closed System Open System Microscopic Level Conclusion Dynamical Fluctuations Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process Conclusions

The MinEP principle Thermodynamic Equilibrium from a Dynamical Fluctuation Law what did we learn in basic courses? Maes, Netoˇ cný After long enough time any closed system is found in one Equilibrium Fluctuations of typical microstates. Closed System Open System Microscopic Level In particular: Conclusion ◮ Macroscopic observables take the values Dynamical Fluctuations Donsker-Varadhan Theory corresponding to the highest number of microstates Equilibrium Dynamics Close-to-Equilibrium = ⇒ Second law of thermodynamics, with entropy MinEP principle Main Result S ( m ) = k B log # { x : m ( x ) = m } Onsager-Machlup Process Conclusions

The MinEP principle Thermodynamic Equilibrium from a Dynamical Fluctuation Law what did we learn in basic courses? Maes, Netoˇ cný After long enough time any closed system is found in one Equilibrium Fluctuations of typical microstates. Closed System Open System Microscopic Level In particular: Conclusion ◮ Macroscopic observables take the values Dynamical Fluctuations Donsker-Varadhan Theory corresponding to the highest number of microstates Equilibrium Dynamics Close-to-Equilibrium = ⇒ Second law of thermodynamics, with entropy MinEP principle Main Result S ( m ) = k B log # { x : m ( x ) = m } Onsager-Machlup Process Conclusions ◮ Macroscopic fluctuations occur with probability Prob ( m ) ∝ e S ( m ) ( k B = 1 ) and for large systems, S ( m ) ≃ Ns ( m ) = ⇒ Einstein’s theory of equilibrium fluctuations

The MinEP principle Equilibrium of Open Systems from a Dynamical Fluctuation Law what did we learn in basic courses? Maes, Netoˇ cný For a system in equilibrium with heat bath at temperature Equilibrium Fluctuations β − 1 , to any macrostate m is assigned the nonequilibrium Closed System free energy Open System Microscopic Level − β − 1 S ( m ) F ( m ) = E ( m ) Conclusion � �� � � �� � Dynamical Fluctuations Energy Entropy Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium ◮ Second law: F ( m ) = min MinEP principle Main Result Onsager-Machlup Process Conclusions

The MinEP principle Equilibrium of Open Systems from a Dynamical Fluctuation Law what did we learn in basic courses? Maes, Netoˇ cný For a system in equilibrium with heat bath at temperature Equilibrium Fluctuations β − 1 , to any macrostate m is assigned the nonequilibrium Closed System free energy Open System Microscopic Level − β − 1 S ( m ) F ( m ) = E ( m ) Conclusion � �� � � �� � Dynamical Fluctuations Energy Entropy Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium ◮ Second law: F ( m ) = min MinEP principle Main Result ◮ Since ∆ S total = ∆ S system + ∆ S bath = − β ∆ F , Onsager-Machlup Process Conclusions (large) equilibrium fluctuations have the law Prob ( m ) ∝ e − β F ( m ) True for large N if F ( m ) ≃ Nf ( m )

The MinEP principle Equilibrium of Open Systems from a Dynamical Fluctuation Law what did we learn in basic courses? Maes, Netoˇ cný For a system in equilibrium with heat bath at temperature Equilibrium Fluctuations β − 1 , to any macrostate m is assigned the nonequilibrium Closed System free energy Open System Microscopic Level − β − 1 S ( m ) F ( m ) = E ( m ) Conclusion � �� � � �� � Dynamical Fluctuations Energy Entropy Donsker-Varadhan Theory Equilibrium Dynamics Close-to-Equilibrium ◮ Second law: F ( m ) = min MinEP principle Main Result ◮ Since ∆ S total = ∆ S system + ∆ S bath = − β ∆ F , Onsager-Machlup Process Conclusions (large) equilibrium fluctuations have the law Prob ( m ) ∝ e − β F ( m ) True for large N if F ( m ) ≃ Nf ( m ) ◮ In specific models, F ( m ) is related to an equilibrium thermodynamic potential

The MinEP principle Equilibrium of Open Systems from a Dynamical Fluctuation Law a microscopic version Maes, Netoˇ cný Equilibrium Fluctuations Closed System Open System ◮ For a system with Hamiltonian H ( x ) , the equilibrium Microscopic Level Conclusion distribution Dynamical Fluctuations Donsker-Varadhan Theory d µ eq ( x ) = ρ eq ( x ) d x = 1 Equilibrium Dynamics Z e − β H ( x ) d x Close-to-Equilibrium MinEP principle Main Result Onsager-Machlup Process Conclusions

The MinEP principle Equilibrium of Open Systems from a Dynamical Fluctuation Law a microscopic version Maes, Netoˇ cný Equilibrium Fluctuations Closed System Open System ◮ For a system with Hamiltonian H ( x ) , the equilibrium Microscopic Level Conclusion distribution Dynamical Fluctuations Donsker-Varadhan Theory d µ eq ( x ) = ρ eq ( x ) d x = 1 Equilibrium Dynamics Z e − β H ( x ) d x Close-to-Equilibrium MinEP principle Main Result is the minimizer of the functional Onsager-Machlup Process Conclusions � H ( x ) ρ ( x ) d x − β − 1 S ( µ ) F ( µ ) = ◮ Gibbs variational principle ◮ S ( µ ) = − R ρ ( x ) log ρ ( x ) d x is the Shannon entropy

The MinEP principle Equilibrium of Open Systems from a Dynamical Fluctuation Law a microscopic version Maes, Netoˇ cný Equilibrium Fluctuations Closed System Open System ◮ For a system with Hamiltonian H ( x ) , the equilibrium Microscopic Level Conclusion distribution Dynamical Fluctuations Donsker-Varadhan Theory d µ eq ( x ) = ρ eq ( x ) d x = 1 Equilibrium Dynamics Z e − β H ( x ) d x Close-to-Equilibrium MinEP principle Main Result is the minimizer of the functional Onsager-Machlup Process Conclusions � H ( x ) ρ ( x ) d x − β − 1 S ( µ ) F ( µ ) = ◮ Gibbs variational principle ◮ S ( µ ) = − R ρ ( x ) log ρ ( x ) d x is the Shannon entropy ◮ F ( ρ ) is related to the fluctuations of empirical distributions!

The MinEP principle Empirical Distributions from a Dynamical Fluctuation Law Maes, Netoˇ cný Construction of the empirical distribution ν x V in a cube V : Equilibrium Fluctuations ◮ For every configuration x consider all translates τ i x for Closed System Open System i ∈ V Microscopic Level Conclusion ◮ Let N x ( y ) be the number of all the translates that Dynamical Fluctuations coincide with y Donsker-Varadhan Theory Equilibrium Dynamics ◮ Define the probability distribution Close-to-Equilibrium MinEP principle Main Result V ( · ) = N x ( · ) Onsager-Machlup Process π x Conclusions | V |

The MinEP principle Empirical Distributions from a Dynamical Fluctuation Law Maes, Netoˇ cný Construction of the empirical distribution ν x V in a cube V : Equilibrium Fluctuations ◮ For every configuration x consider all translates τ i x for Closed System Open System i ∈ V Microscopic Level Conclusion ◮ Let N x ( y ) be the number of all the translates that Dynamical Fluctuations coincide with y Donsker-Varadhan Theory Equilibrium Dynamics ◮ Define the probability distribution Close-to-Equilibrium MinEP principle Main Result V ( · ) = N x ( · ) Onsager-Machlup Process π x Conclusions | V | ⇒ π x → µ eq , for any typical x Idea : Spatical ergodicity =