The Problem of Constructing Phenomenological Equations for Subsystem Interacting with non-Gaussian Thermal Bath

1
The Problem of Constructing Phenomenological
Equations for Subsystem Interacting with
non-Gaussian Thermal Bath
UPoN 2008 Lyon (France), June 2-6

• Alexander Dubkov

Nizhniy Novgorod State University, Russia
Peter Hänggi and Igor Goychuk
Institut für Physik, Universität Augsburg, Germany
This work was supported by RFBR grant 08-02-01259
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The Problem of Constructing Phenomenological
Equations for SubsystemInteracting with
non-Gaussian Thermal Bath
OUTLINE

• Introduction
• Different methods to obtain a stochastic Langevin
  equations with Gaussian thermal bath
• Constructing the Langevin equation for Brownian
  particle interacting with non-Gaussian thermal
  bath
• Additive or multiplicative noise? Stratonovichs
  approach to constructing Fokker-Planck equations
• Conclusions

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The Problem of Constructing Phenomenological
Equations for SubsystemInteracting with
non-Gaussian Thermal Bath
1. Introduction
The main problem of phenomenological
theory construction of the thermodynamically
correct stochastic equations for variables of
subsystem interacting with thermal bath
CENTRAL LIMIT THEOREM ? GAUSSIAN THERMAL
BATH
Classic Langevin equation with white Gaussian
random force
Einsteins relation
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The Problem of Constructing Phenomenological
Equations for SubsystemInteracting with
non-Gaussian Thermal Bath
Generalized Langevin equation (GLE) of
Kubo-Mori type
Fluctuation-dissipation theorem
Why non-Gaussian thermal bath?

• Particle collisions with molecules of solvent
  (Poissonian noise)
• Electrical circuits with nonlinear resistance at
  thermal equilibrium
• A relatively small number of charge carriers in a
  conductors
• Anharmonic molecular vibrations in molecular
  solids
• Newtons nonlinear friction (?(v)v)

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The Problem of Constructing Phenomenological
Equations for SubsystemInteracting with
non-Gaussian Thermal Bath
Experimental evidence of non-Gaussian statistics
of current fluctuations in thin metal films at
thermal equilibrium R.F.Voss and J.Clarke, Phys.
Rev. Lett. 1976. V.36. P.42 Phys. Rev. A 1976.
V.13. P.556
Theoretical investigations
A knowledge of nonlinear dissipative flow is not
sufficient to reconstruct the original stochastic
dynamics P. Hänggi, Phys. Rev. A 1982. V.25.
P.1130
Derivation of the current-voltage characteristic
of the semiconductor diode from Poissonian model
of charge transport G.N. Bochkov and A.L. Orlov,
Radiophys. and Quant. Electron. 1986. V.29. P.888
Nonlinear stochastic models of oscillator
systems G.N. Bochkov and Yu.E. Kuzovlev,
Radiophys. and Quant. Electron. 1976. V.21. P.1019
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The Problem of Constructing Phenomenological
Equations for SubsystemInteracting with
non-Gaussian Thermal Bath
2. Different methods to reconstruct stochastic
macrodynamics
Excluding thermal bath variables from microscopic
dynamics (Kubo approach, Rep. Progr. Phys. 1966,
V.29, P.255)
Equations of Hamiltonian mechanics
?
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The Problem of Constructing Phenomenological
Equations for SubsystemInteracting with
non-Gaussian Thermal Bath
After solving the second equation and
substituting in the first one we find
Because of
we immediately obtain GLE and fluctuation-dissipat
ion theorem
Phenomenological approach
Equilibrium Gibbsian distribution
?
Basic principles of statistical mechanics
?
Microscopic time reversibility
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The Problem of Constructing Phenomenological
Equations for SubsystemInteracting with
non-Gaussian Thermal Bath
We will try to describe the particle of mass m
interacting with non-Gaussian white thermal bath
?(t) of the temperature T by the Langevin
equation containing additive noise source
where ?(v) is unknown nonlinear dissipation
We use the general Kolmogorovs equation obtained
in the paper A. Dubkov and B. Spagnolo, Fluct.
Noise Lett. 2005. V.5, P.L267
Taking into account the evident condition ?(0) 0
we find in asymptotics
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The Problem of Constructing Phenomenological
Equations for SubsystemInteracting with
non-Gaussian Thermal Bath
where
is the equilibrium Maxwell distribution
If the moments of the kernel function are finite
we arrive at
where
are Hermitian polynomials
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The Problem of Constructing Phenomenological
Equations for SubsystemInteracting with
non-Gaussian Thermal Bath
For white Gaussian noise source ?(z)2D?(z)
we have
For Poissonian white noise
with Gaussian distribution of amplitudes
is the error function
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The Problem of Constructing Phenomenological
Equations for SubsystemInteracting with
non-Gaussian Thermal Bath
3. Additive of multiplicative?
In accordance with Stratonovichs theory R.L.
Stratonovich, Nonlinear Nonequilibrium
Thermodynamics. Springer-Verlag, Berlin, 1992
the stochastic Langevin equation should be
multiplicative!
From Kolmogorovs equation we find for such a case
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The Problem of Constructing Phenomenological
Equations for SubsystemInteracting with
non-Gaussian Thermal Bath
If we put ?P/ ?t0 we obtain complex relation
between three functions
is the Maxwell equilibrium distribution
where
Choosing the statistics of noise ?(z) we have the
relationship between the nonlinear dissipation
and velocity-dependent noise intensity
For Poissonian noise we arrive at
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The Problem of Constructing Phenomenological
Equations for SubsystemInteracting with
non-Gaussian Thermal Bath
Conclusions
1. For additive noise the nonlinear friction
function can be derived exactly for given
statistics of the thermal bath. 2. The
construction of physically correcting stochastic
Langevin equation corresponding to the
non-Gaussian thermal bath and the nonlinear
friction requests introducing a multiplicative
noise source. 3. This noise source should be
non-Gaussian. 4. Using the Gibbsian form of the
equilibrium distribution one can find only
relation between the nonlinear friction and the
velocity-dependent noise intensity. 5. Solution
of this unsolved problem in the noise theory
opens a way to the new realm of Brownian motion.
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