1. Ｅｎｅｒｇｙ dissipation and FDR violation
Shin-ichi Sasa (Tokyo)
Paris, 2006, 09/2８

2. Introduction Fluctuation-dissipation relation (FDR)
a fundamental relation in linear response theory
Violation in systems far from equilibrium
There is a certain universality in a manner of the violation : e.g. Effective temperature
Cugliandolo, Kurchan, and Peliti, PR E, 1997
Berthier and Barrat, PRL, 2002

3. Content of this talk For a class of Langevin equations
　 describing a nonequilibrium steady state,
the violation of FDR is connected to the energy dissipation ratio as an equality.
Ref. Harada and Sasa, PRL in press;
cond-mat/
A microscopic description of the equality
Ref. Teramoto and Sasa, cond-mat/

4. A simple example
periodic boundary condition

5. Quantities Statistical average under the influence of the probe force
Stratonovich interpretation
The energy interpretation was given by Sekimoto in 1997

6. Theorem stationarity Equilibrium case (no external driving)
Fluctuation-dissipation
Theorem (FDT)

7. Ｑｕｉｃｋ　ｄｅｒｉｖａｔｉｏｎ

9. Remark (generalization to..)
Many body Langevin system
Langevin system with a mass term
Langeivn system with time-dependent potential (e.g. stochastic, periodic)
Langevin system with multiple heat reservoirs
Ref. Harada and Sasa, in preparation
cond-mat/0910***

10. Significance
The equality is closed with experimentally measurable quantities
The equality does not depend on the details of the system (e.g. potential functions)
The equality connects the kinematic quantities (correlation and response functions) with the energetic quantity (energy dissipation ratio).
Universal statistical property related to energetics .

11. Micrsoscopic description
potential
driving force

12. Equation of motion: Hamiltonian equation
Bulk-driven Hamiltonian
system; H involves the
potential
Temperature control only
at the boundaries by the
Nose-Hoover method

13. Distribution function
time-dependent distribution function
Evolution equation:
Initial condition:
= the stationary solution for the system
Why this choice ? I will be back later.

14. Solving the equation
Set
We can solve this equation formally as

15. Solution
Zubarev-McLennan
type expression

16. FDR violation: exact expression

17. Physical consideration
Time scale of (B,Y,λ）　
Time scale of V
Time scale of R

18. Remark (generalization to)
Sheared systems
Elerctric (heat) conduction systems
….. not yet
A formal exact expression of FDR violation
is always obtained, but not useful in general.
(Remember the choice of the initial condition
in the simple example discussed above.)

19. Distribution function II
time-dependent distribution function
Initial condition:
= the stationary solution for the system
The same steady distribution in the limit
Different expression of the FDR violation
difficult to connect it to the result for the Langevin

20. Slow relaxation system
: initial values are sampled randomly
: a magnetic field is turned on

21. A formal result Relation to energy relaxation ?
effective temperature ?
cf. Cugliandolo, Dean, Kurchan, PRL, 1997

22. Summary
I presented an equality connecting FDR violation with energy dissipation.
I provided a proof of this equality.
I described this equality on the basis of microscopic dynamics.
Toward a useful characterization of statistical
properties in terms of energetic quantities
for a wide class of non-equilibrium systems.

23. Question 1
Q: The energy dissipation can be discussed by using response function in linear response theory. In there a relation with this?
A:We do not find a clear direct relation with linear response theory, but both the theories are correct and compatible. Note that the response function in linear response theory is defined as that to an equilibrium state, not to a steady state.

24. Question 2
Q: Your argument neglects the hydrodynamic effect. Is it possible to take this effect into account?
A: It will be possible, but not yet done. In a microscopic description, this incorporation is more difficult than to study simple shear flow. Thus, the priority is not the first. If you wish to analyze a phenomenological description of the Brownian particles with the hydrodynamic effect, you can calculate an expression of the FDR violation for this model. It might be interesting to investigate the expression from an energetic viewpoint.

25. Question 3
Q : Can your analysis be applied to the other thermostat models ?
A: No. For example, there is a technical problem to analyze a system with a Langevin type thermostat at boundaries. However, I expect that this is not essential and will be solved soon.

26. Question 4
Q Is it possible to analyze a pure Hamiltonian system without thermostat walls ?
A Yes, if you do not take care of the mathematical rigor,
but the rigorous mathematical treatment is challenging.