1. Hisao Hayakawa (YITP, Kyoto University) Physics and chemistry in quantum dissipative systems, YITP, Kyoto University(2010/08/09) Fluctuation theorem and generalized Green-Kubo formula for quantum dissipative systems

2. Contents Introduction General framework of classical and quantum system Fluctuation theorem and generalized Green-Kubo formula A simple example Summary References: Chong et al, PRE 81, 041301 (2010); HH, PTP Suppl. No.184, 545 (2010) and to be submitted.

3. Introduction Purpose of this talk

4. I am studying granular physics. Each grain is large and the heat can be absorbed in it. The grain is coupled with the heat bath at T=0. No detailed balance T=0 Nonequilibrum steady state (NESS) under an external force Work Dissipation

5. Nonequilibrium steady state T, ρ(0) T_{ss}, ρ_{SS} External force dissipation

6. Quantum nonequilibrium physics We formulate physics of nonequilibrium steady state for granular physics. I realize that the general framework is common even in both quantum dissipative case and classical liquid theory. We can revisit the fluctuation theorem and Green-Kubo formula as results of identities. system environment

7. Purpose of this talk Construction of a general framework of dissipative system from the Liouville equation. Fluctuation theorem involves Green-Kubo formula. ◦FT is equivalent to Jarzynski equality. ◦FT involves Onsager’s reciprocal relations (and FDR). We should clarify the role of FT including the role of dissipation.

8. Fluctuation theorem Green-Kubo FDR

9. Part II General framework of classical and quantum dissipative systems

10. General framework We begin with a general framework without loss of any information. We can apply this approach to any dissipative systems both classical and quantum systems. ◦See the framework by Evans & Morriss The results are exact but general nonsense.

11. Part II-A General framework for classical systems

12. Time evolution of a classical system Phase variables: positions and momenta Time evolution of a phase variable The matrix for the time evolution satisfies

13. Distribution function Continuity equation Liouvillian is not unitary. Λ is the phase volume contraction. Time evolution of the distribution function satisfies Phase volume

14. Time evolution matrices

15. The average is independent of its picture. Any two Liouville operators satisfy the Dyson equation

16. Kawasaki representation Kawasaki representation is an identity between two time evolution matrices (Kawasaki and Gunton, 1973). From Kawasaki representation and independence of representation leads to

17. Part II-B General framework for qunatum systems

18. Framework for quantum case. General framework is common even for quantum case. We assume that the observable A satisfies This means that we restrict our interest to the case of factorizing initial condition.

19. Liouville equation The subsystem is obtained from the trace of the environment. Liouville equation is given by

20. Time evolution of density matrix Time evolution is given by where The subsystem satisfies

21. Dissipator Dissipator Nakajima-Zwanzig model Lindblad model

22. Adjoint dynamics The adjoint dynamics is given by Mori equation Lindblad

23. Trace and Dyson equation The average value is given by with Dyson equation is unchanged

24. Part III Fluctuation theorem and generalized Green-Kubo for dissipative systems

25. Kawasaki representation Kawasaki representation can be used even for quantum case. From Kawasaki representation and independence of representation leads to if we start from the canonical initial condition (or density matrix)

26. Integral FT and 2 nd law The meaning of integral FT These represent a generalized 2nd law of thermodynamics. Jansen’s inequality

27. Derivation of the integral fluctuation theorem The fluctuation theorem is the consequence of Kawasaki’s representation. This is obtained by setting A=1.

28. Generalized Green-Kubo formula for classical case Time differentiation of Integral fluctuation theorem at finite t and integate it over t. Then we obtain in the limit t->∞ For classical shear stress we obatin where

29. Dissipation free systems If we omit the dissipation for a uniform shear system, we obtain known Green-Kubo formula for stress This is because in the dissipationless case.

30. Generalized Green-Kubo formula for quantum systems

31. Generalized Green-Kubo (2) where we have used Kubo’s identity. The density matrix is thus given by j_H(t) evolves with the full Hamiltonian.

32. Part IV A simple example: results for Calderia- Leggett model

33. An example for Calderia-Leggett model

34. Master equation

35. Density matrix

36. Generalized Green-Kubo formula

37. A simple example This is the trivial example with V(x)=0. In this case, the conventional Green-Kubo formula gives the exact result with

38. Summary So far, we have not used any specific properties both quantum and classical systems. Therefore, our results can be used for any dissipative system for the factorizing initial condition. Fluctuation theorem is the universal law to govern the dissipative systems.