1. TEST OF THE GALLAVOTTI-COHEN SYMMETRY IN A STOCHASTIC MODEL WITH NON EQUILIBRIUM STATIONARY STATES Giuseppe Gonnella Antonio Lamura Antonio Piscitelli

2. Equilibrium stationary statesGibbs-Boltzmann distribution Non equilibrium stationary states (N.E.S.S.) -Thermal gradient -Energy flow imposed by the extern 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 to E.G.D.Cohen, R.van Zon PHYS.REV.E 69,056121

3. GALLAVOTTI-COHEN SYMMETRY FOR N.E.S.S. Ergodic non equilibrium stationary states: Trajectory in phase space Total energy injected into the system, work done by the extern, over a time interval Average of the average values ofover the subsequent intervals of timealong the history Poniamo e

4. The theorem suggests that: in a non equilibrium stationary state the probability distribution function where satisfies that Probability distribution function that the ratio is called Symmetry Function assumes the valuein the time interval

5. TEST OF GALLAVOTTI-COHEN SYMMETRY IN STOCHASTIC LANGEVIN SYSTEM FOR BINARY MIXTURES The model: = order parameter Evolution equation: with: (Noise verifies the fluctuation-dissipation relation) and 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.

6. POWER DEFINITION IN SYSTEMS WITH SHEAR = Pressure tensor Power density= (De Groot-Mazur) Non diagonal part of pressure tensor = (A.J.M.Yang, P.D.Fleming, J.H.Gibbs, Journal of Chemical Physics, vol.64, No.9)

7. x y shear direction x y direction opposite to that of the shear Probability distribution function for

8. We work above the critical temperature Remember thatis 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

9. Temporal correlation of Correlation of stress GENERAL BEHAVIOUR OF THE SYSTEM Configuration at a stationary time time spent by C(X) for reaching at 95% its stationary value.

10. SOME RESULTS

13. SLOPES

14. CONCLUSIONS the limit slope (1 for GC symmetry) seems to vary with and 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

15. Thank you for the attention!