2. Thermodynamics, fluctuations, and response for systems out of equilibrium Shin-ichi Sasa (University of Tokyo) 2007/11/05 in collaboration with T.S. Komatsu, N. Nakagawa, and H. Tasaki

3. Outline of my talk 1.Introduction (7min) 2.Question (4min) 3. Result (14min) 4. Conclusion (1min) 2

4. (Near and in) Equilibrium 3 Entropy Thermodynamic function Second law Fundamental limitation of operations Thermodynamic relation Unified description of material properties dU=TdS-pdV Large deviation Macroscopic fluctuations Equilibrium distribution expression in terms of “energetic quantities” Fluctuation-dissipation relation Linear response formula Detailed-balance Microscopic reversibility The principle of equal weightMicroscopic equation

5. Non-equilibrium steady state 4 Entropy Thermodynamic function Second law Fundamental limitation of operations Thermodynamic relation Unified description of material properties dU=TdS-pdV Large deviation Macroscopic fluctuations Stationary distribution expression in terms of “energetic quantities” Violation of Fluctuation-dissipation relation Detailed-balance Microscopic reversibility The principle of equal weightMicroscopic equation ？ ？ ？ X ？

6. Our results 5 Entropy Second law Fundamental limitation of operations Thermodynamic relation Unified description of material properties Large deviation Macroscopic fluctuations Stationary distribution expression in terms of “energetic quantities” A formula for the violation of FDR Local detailed-balance Microscopic reversibility Microscopic equation (Hatano-Sasa, PRL, 2001) (Sasa-Tasaki, JSP, 2006) (Harada-Sasa, PRL, 2005)(Sasa, arXiv0706.0043 ) (Komatsu-Nakagawa, arXiv0708.3158 ) Valid up to O(     : the degree of non-equilibrium (KNST, arXiv0711.0426) (KNST, 2007) (KNST=Komatsu, Nakagawa, Sasa and Tasaki)

7. Outline of my talk 1.Introduction (7min) 2.Question (4min) 3. Result (14min) 4. Conclusion (1min) 6

8. Heat conducting steady state 7 Heat bath systemHeat bath (volume) Hamiltonian system Stochastic system (Hamiltonian system) and other ….. Stochastic system (Hamiltonian system) and other ….. piston Set of parameters in the Hamiltonian Microscopic description Control parameter

9. Equilibrium case 8

10. Question Existence of “Thermodynamic function” F(T  in non-equilibrium steady state ? 9 - These should be determined operationally. - It should contain new predictions that can be checked experimentally T “temperature” Y characterizes the non-equilibrium nature

11. difficulty To seek for such a framework is a danger project, because there is much ambiguity. “What is T? “ “ What is Y? “ Careful arguments (with proposing a possible form) were presented in the paper, Sasa and Tasaki, JSP, 2006 (100 pages!)

12. Outline of my talk 1.Introduction (7min) 2.Question (4min) 3. Result (14min) 4. Conclusion (1min) 11

13. Heat conducting steady state 12 Heat bath systemHeat bath (volume) Hamiltonian system Stochastic system (Hamiltonian system) and other ….. Stochastic system (Hamiltonian system) and other ….. piston Set of parameters in the Hamiltonian Microscopic description Control parameter

14. Useful representation 13 Conditional path ensemble average  arXiv0708.3158Komatsu-Nakagawa Energy into the left heat bath Energy into the right heat bath dimensionless heat flux

15. Remark 14 Equilibrium case S : thermodynamic entropy

16. Significance 15 - Linear response formula (in an elegant manner) - Universal form for many non-equilibrium systems - Non-linear response formula - Large deviation functional - Steady state thermodynamics

17. Clausius’ formula 16 Quasi-static protocol Equilibrium case Quasi-static heat out of the baths Non-equilibrium case

18. Extended Clausius’ formula 17 (KNST, preprint, arXiv:0711.0426) Quasi-static reverse protocol

19. Thermodynamics 18 Heat bath systemHeat bath (volume) Hamiltonian system Stochastic system (Hamiltonian system) and other ….. Stochastic system (Hamiltonian system) and other ….. (KNST, in preparation)

20. Result 19 Existence of Steady State Thermodynamics ! (KNST, preprint, arXiv:0711.0246) The bulk region is changed only in the J direction pressure in the J direction e.g.

21. Significance I Universal macroscopic theory based on microscopic mechanics ! Non-trivial relation among experimentally measurable quantities (through the Maxwell relation) All theories that discuss statistical behavior beyond the linear response regime must satisfy our relation e.g. Compressibility tensor in a heat conducting fluid e.g. Check the validity of the Boltzmann equation

22. Significance II New physics in terms of Mechanical detection of the departure from the local equilibrium (such as “long range correlation”) Enlightening guide in a (future) construction of non- equilibrium statistical mechanics ! 21

23. Summary 22 Entropy Second law Fundamental limitation of operations Thermodynamic relation Unified description of material properties Large deviation Macroscopic fluctuations Stationary distribution expression in terms of “energetic quantities” A formula for the violation of FDR Local detailed-balance Microscopic reversibility Microscopic equation (Hatano-Sasa, PRL, 2001) (Sasa-Tasaki, JSP, 2006) (Harada-Sasa, PRL, 2005)(Sasa, arXiv0706.0043 ) (Komatsu-Nakagawa, arXiv0708.3158 ) Valid up to O(     : the degree of non-equilibrium (KNST, arXiv0711.0426) (KNST, 2007) (KNST=Komatsu, Nakagawa, Sasa and Tasaki)