test of the gallavotti-cohen symmetry in a stochastic model with non equilibrium stationary states
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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!