TEST OF THE TEST OF THE GALLAVOTTI-COHEN GALLAVOTTI-COHEN SYMMETRY IN A SYMMETRY IN A STOCHASTIC MODEL STOCHASTIC MODEL WITH NON WITH NON EQUILIBRIUM EQUILIBRIUM STATIONARY STATESSTATIONARY STATES

Giuseppe GonnellaGiuseppe GonnellaAntonio LamuraAntonio LamuraAntonio Piscitelli Antonio Piscitelli

Equilibrium stationary states Gibbs-Boltzmann distribution

Non equilibrium stationary states (N.E.S.S.)

-Thermal gradient

-Energy flow imposed by the extern

)( *tvxkF tt

'*'dtFvWt

t

t

t t

Single brownian particle dragged through water by a laser induced moving potential

Force exerted on the particle

Work done on the system over a time interval from

toE.G.D.Cohen, R.van Zon

PHYS.REV.E

69,056121

GALLAVOTTI-COHEN SYMMETRY FOR N.E.S.S.

Ergodic non equilibrium stationary states:

Trajectory in phase space

W Total energy injected into the system, work done by the extern, over a time interval

WAverage of the average values of W over the subsequent intervals of time along the history

Poniamo

Ww

w

W

W

Wp e

The theorem suggests that:

in a non equilibrium stationary state the probability distribution function

ppfW

);(lim

);(

);(ln

1);(

p

p

wpfW

where

satisfies that

);( p Probability distribution function that the ratio

is called Symmetry Function

W

W

assumes the value p in the time interval

TEST OF GALLAVOTTI-COHEN SYMMETRY IN STOCHASTIC LANGEVIN SYSTEM FOR BINARY MIXTURES

The model:

),( tx = order parameter

),(2 txF

vt

Evolution equation:

with: 0),( tx )()(2),(),( ''2'' ttxxTtxtx

(Noise verifies the fluctuation-dissipation relation)

and

242

242

kgrxdF d 0,, kgr

0)0,( x )()0,(),0,( '' xxxx

NOTE: This model is used in practise in the quite general framework of the study of phase separation (with r<0) and of mixtures dynamics with a convective term, when the fluctuations of the velocity field are negligible.

POWER DEFINITION IN SYSTEMS WITH SHEAR

= Pressure tensor

Power density=xyjiij

v (De Groot-Mazur)

Non diagonal part of pressure tensor = yxxy k (A.J.M.Yang, P.D.Fleming, J.H.Gibbs,

Journal of Chemical Physics, vol.64, No.9)

)()( yxkdxdy

)(P

)ln()ln()ln(

77.0

0 yxxy

kP

x

y

shear direction

0

yxxykP

x

y

direction opposite to that of the shear

W

2.07

3000Nbin

Probability distribution function for

We work above the critical temperature )(CTT

Remember that )(CT is an increasing function of .

Can be found more on this topic on

F.Corberi,G.Gonnella,E.Lippiello,M.Zannetti,

J.Phys.A:Math.Gen. 36 No 17

4729-4755

Temporal correlation of Correlation of stress

GENERAL BEHAVIOUR OF THE SYSTEM

Configuration at a stationary time

34.0

)(C

)(C

25.0

1.05108

time spent by C(X) for reaching at 95% its stationary value.

:X

5.01,1 gr

SOME

RESULTS

W

W

005.036.1 slope

W

Wf

W

2000Nbin5.0

05.0

05.010

W

W

W

Wf

019.086.0 slopeW

2000Nbin

2.05.0

W

034.075.2 slope W

W

Wf

W

3000Nbin

SLOPES05.0 1.0

2.0

CONCLUSIONS

the limit slope (1 for GC symmetry) seems to vary withand in particular seems to increase with

It has been measured the correlation time (of and ) and it has been verified that the characteristics and the behaviour of the system are typical of a stationary state above the critical temperature.

From the simulations performed until now

NEXT STEPS

To state with more precision the above result for understanding better the trend of the limit slope with

To approach the problem analytically

Thank you for the attention!