speaker:Dario Villamaina

Sapienza University, Rome

In collaboration with:Andrea Baldassarri

Andrea PuglisiAngelo Vulpiani

Restoring fluctuation-dissipation relations in non equilibrium steady states

Dresden, Max Planck Institut 2009

1. Correlation-response relations in steady states

3. Numerical analysis of response on granular gas models

Structure of the talk

2. Two temperatures brownian motion (exactly solvable model)

Classical fluctuation-dissipation relations

Einstein relation

Kubo relation

Equivalent to the integrated version

Mobility(integrated response)

Diffusion coefficient(integrated correlation)

Valid for a Brownian particle in a fluid at temperature T

Valid for Hamiltonian systems at temperature with a variation

It also holds for a Langevin equation with a gradient structure

Generalized Fluctuation-dissipation relation (GFDR)

Perturbed trajectory:

Steady state correlation(unperturbed system)

Relaxation of anexternal perturbation

if is a non vanishing and differentiable phase space distribution function and the system is mixing

GFDRKubo relation

Einstein relation

Evolution operator

is quadratic if

if

M. Falcioni, S. Isola, A. Vulpiani - Physics letters. A, 1990UMB. Marconi, A. Puglisi, L. Rondoni, A. Vulpiani – Physics Report 2008.

2. Two temperatures brownian motion (exactly solvable model)

Brownian motion with memory: two thermostats case

Fast thermostat (white noise) Slow thermostat (red noise)

Stochastic partMemory termHarmonic force

Brownian motion with memory:Equilibrium case (Tfs=T

sf)

Free particle case: k=0 Overdamped regime

Equilibrium case: and

Distribution of velocity is always a gaussian with variance Teff:

Is this a violation of GFDR?

k=0 case Overdamped case

Brownian motion with memory: two thermostats case (Tf=T

s)

F. Zamponi, F. Bonetto, LF Cugliandolo, J Kurchan - J. Stat Mech, 2005

Memory as a hidden variable

Colored noise and memory term are not present

Mapped in a two variables system

Memory as a hidden variable

Two variables Langevin equationMemory is a degree of freedom

ρ(u,v) has a bivariate Gaussian shape:

Application of GFDR

observed is a marginal distribution

How do you know you have taken enough variables, for it to be markovian?Lars Onsager

Considering the “missing” variable linearity is restored

Restoring FDR D. Villamaina, A. Baldassarri, A. Puglisi, A. Vulpiani – to be published

Similar considerations are also valid:in the overdamped regime

when several thermostat are present

Non linear relations between Cvv(t) and Rvv(t)

(

are due to the cross correlation term

This is not a violation of generalized FDR!

Coupling effect vanishes at equilibrium

3. Numerical analysis of response on granular gas models

Two times scale system Mean collision time

Thermostat time τb

τc

Granular gas model:Puglisi, Baldassarri, Loreto - Physical Review E, 2002

Equilibrium-like regime τc>>τ

bColliding regime τ

c<τ

b

Homogeneous spatial distributionMaxwell distribution of velocities

Spatial inhomogeneity is presentNon-Maxwellian deviations can occur

Response Autocorrelation

Granular Temperature

Einstein relation

Perturbation of tracer's velocity(linear response regime)

Dilute regime:

Pdf is well fitted by One may expect

Surprisingly (very small deviations are observed)

A. Puglisi, A Baldassarri, A Vulpiani – J. Stat. Mech, 2007D. Villamaina, A. Puglisi, A Vulpiani - J. Stat. Mech, 2008

Coupling beetween different degrees of freedom producesnon linear relations between C(t) and R(t)

b

Strong dissipation

In a strong dissipation regime Einstein relation is not recoveredThis is not a violation of

Fluctuaction-dissipationRelations (FDR)

R

Generalized FDR

Spatial inhomogeneity Single particle factorization fails

Packing fraction

Simple ansatz on introducing a local velocity average field u(x)

S

Restoring equilibrium-like relations

The generalized FDR can be used to verify hypothesis on the phase space distribution function

u(x) is defined on a small cell of diameter centered in the tracer

A pedagogical mistakeElastic regime

A misleading analysis of RC plot could cause paradoxes (Two effective temperatures in the elastic case!)

Actually it is an effect of the non-trivial velocity correlation’s shape

Strong inelastic regime

Conclusions

In non equilibrium steady states, classical fluctuation-dissipation relations are non expected to hold

It is possible to define a generalized formula relating response function to a correlation computed in the steady state

A non linear relation between autocorrelation and response functions is due to couplings between different degrees of freedom

The generalized relation can be used to verify onsets on the steady state distribution function

References

Paper we published on GFDR in granular gas models:

Recent works on generalized fluctuation dissipation relation :

A. Puglisi, A. Baldassarri, A. Vulpiani, J. Stat. Mech. (2007)

D. Villamaina, A. Puglisi, A. Vulpiani, J. Stat. Mech. (2008)

U. Marini Bettolo Marconi, A. Puglisi, L. Rondoni, A. Vulpiani, Physics Report (2008)

D. Villamaina, A. Baldassarri, A. Puglisi, A. Vulpiani, to be published

Numerical recipes for FDR computation

Tthe gas is left evolve from an initial condition until a statistically stationary state is reached

Simulation are performed in a linear response regime

A copy of the system is obtained, identical to the original except one particle, whose velocity’s component is augmented of

Both the systems are left evolve with the unperturbed dynamic (same noise is used)