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

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

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/0502505 A microscopic description of the equality Ref. Teramoto and Sasa, cond-mat/0509465

A simple example periodic boundary condition

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

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

Ｑｕｉｃｋ　ｄｅｒｉｖａｔｉｏｎ

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***

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 .

Micrsoscopic description potential driving force

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

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

Solving the equation Set We can solve this equation formally as

Solution Zubarev-McLennan type expression

FDR violation: exact expression

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

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.)

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

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

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

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.

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.

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.

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.

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.