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TEST OF THE GALLAVOTTI-COHEN SYMMETRY IN A STOCHASTIC MODEL WITH NON EQUILIBRIUM STATIONARY STATES. Giuseppe Gonnella Antonio Lamura Antonio Piscitelli. Equilibrium stationary states. Gibbs-Boltzmann distribution. Non equilibrium stationary states (N.E.S.S.). -Thermal gradient
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Non equilibrium stationary states (N.E.S.S.)
-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
E.G.D.Cohen, R.van Zon
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 of
over the subsequent
intervals of time
along the history
assumes the value
in the time interval
The theorem suggests that:
in a non equilibrium stationary state the probability distribution function
is called Symmetry Function
TEST OF GALLAVOTTI-COHEN SYMMETRY IN STOCHASTIC LANGEVIN SYSTEM FOR BINARY MIXTURES
= order parameter
(Noise verifies the fluctuation-dissipation relation)
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.
= Pressure tensor
Non diagonal part of pressure tensor =
(A.J.M.Yang, P.D.Fleming, J.H.Gibbs,
Journal of Chemical Physics, vol.64, No.9)
direction opposite to that of the shear
is an increasing function of .
Can be found more on this topic on
J.Phys.A:Math.Gen. 36 No 17
Temporal correlation of
Correlation of stress
time spent by C(X) for reaching at 95% its stationary value.
Configuration at a stationary time
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
the limit slope (1 for GC symmetry) seems to vary with
and in particular seems to increase with
To state with more precision the above result for understanding better the trend of the limit slope with
To approach the problem analytically