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

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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

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Equilibrium stationary states
Gibbs-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
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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 of
over the subsequent
intervals of time
along the history
e
Poniamo
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Probability distribution function that the ratio
assumes the value
in the time interval
The theorem suggests that
in a non equilibrium stationary state the
probability distribution function
satisfies that
where
is called Symmetry Function
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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 rlt0) and of mixtures dynamics
with a convective term, when the fluctuations of
the velocity field are negligible.
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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)
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Probability distribution function for
shear direction
direction opposite to that of the shear
y
y
x
x
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We work above the critical temperature
Remember that
is an increasing function of .
Can be found more on this topic on
F.Corberi,G.Gonnella,E.Lippiello,M.Zannetti, J.Phy
s.AMath.Gen. 36 No 17 4729-4755
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GENERAL BEHAVIOUR OF THE SYSTEM
Temporal correlation of
Correlation of stress
time spent by C(X) for reaching at 95 its
stationary value.
Configuration at a stationary time
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SOME RESULTS
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SLOPES
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CONCLUSIONS
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
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
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Thank you for the attention!