The Problem of Constructing Phenomenological Equations for Subsystem Interacting with non-Gaussian Thermal Bath Alexander Dubkov Nizhniy Novgorod State

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

Alexander DubkovAlexander Dubkov

Nizhniy Novgorod State University, Russia

UPoN 2008UPoN 2008

Lyon (France),Lyon (France), JuneJune 2-62-6

This work was supported by RFBR grant 08-02-01259

Peter HPeter Häänggi and Igor Goychuknggi and Igor Goychuk

Institut für Physik, Universität Augsburg, Germany

OUTLINEOUTLINE

• IntroductionIntroduction

• Different methods to obtain a stochastic Langevin Different methods to obtain a stochastic Langevin equations with Gaussian thermal bathequations with Gaussian thermal bath

• Constructing the Langevin equation for Brownian Constructing the Langevin equation for Brownian particle interacting with non-Gaussian thermal bathparticle interacting with non-Gaussian thermal bath

• Additive or multiplicative noise? Stratonovich’s Additive or multiplicative noise? Stratonovich’s approach to constructing Fokker-Planck equationsapproach to constructing Fokker-Planck equations

• ConclusionsConclusions

The Problem of Constructing Phenomenological Equations for SubsystemThe Problem of Constructing Phenomenological Equations for SubsystemInteracting with non-Gaussian Thermal BathInteracting with non-Gaussian Thermal Bath

UPoN 2008UPoN 2008 Lyon (France),Lyon (France), June 2-6June 2-6 22



1. Introduction1. 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

DtFtFtFtFvdt

dvm 2)()(,0)(),(

Einstein’s relationEinstein’s relation Tk

D

B

Classic Langevin equation with white Gaussian random forceClassic Langevin equation with white Gaussian random force



Why non-Gaussian thermal bath?Why non-Gaussian thermal bath?

• Particle collisions with molecules of solvent (Particle collisions with molecules of solvent (PoissonianPoissonian noise) noise)

• Electrical circuits with nonlinear resistance at thermal equilibriumElectrical circuits with nonlinear resistance at thermal equilibrium

• A relatively small number of charge carriers in a conductorsA relatively small number of charge carriers in a conductors

• Anharmonic Anharmonic molecular vibrations in molecular solidsmolecular vibrations in molecular solids

• Newton’s nonlinear friction (Newton’s nonlinear friction ((v)~|v|(v)~|v|))

Generalized Langevin equation (GLE) of Kubo-Mori typeGeneralized Langevin equation (GLE) of Kubo-Mori type

Ft

KtFtFtFtFdvtdt

dvm )()(,0)(),(

Fluctuation-dissipation theorem:Fluctuation-dissipation theorem: 0, TkK BF



Derivation of the current-voltage Derivation of the current-voltage characteristic of the semiconductor characteristic of the semiconductor

diode from Poissonian model of diode from Poissonian model of charge transportcharge transport

G.N. Bochkov and A.L. Orlov, Radiophys. and G.N. Bochkov and A.L. Orlov, Radiophys. and Quant. Electron. 1986. V.29. P.888Quant. Electron. 1986. V.29. P.888

A knowledge of nonlinear dissipative flow is not sufficient to A knowledge of nonlinear dissipative flow is not sufficient to reconstruct the original stochastic dynamicsreconstruct the original stochastic dynamics

P. HP. Häänggi, Phys. Rev. A 1982. V.25. P.1130nggi, Phys. Rev. A 1982. V.25. P.1130

Nonlinear stochastic models of Nonlinear stochastic models of oscillator systemsoscillator systems

G.N. Bochkov and Yu.E. Kuzovlev, Radiophys. G.N. Bochkov and Yu.E. Kuzovlev, Radiophys. and Quant. Electron. 1976. V.21. P.1019and Quant. Electron. 1976. V.21. P.1019

Experimental evidence of non-Gaussian statistics of current Experimental evidence of non-Gaussian statistics of current fluctuations in thin metal films at thermal equilibriumfluctuations 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.556R.F.Voss and J.Clarke, Phys. Rev. Lett. 1976. V.36. P.42; Phys. Rev. A 1976. V.13. P.556

Theoretical investigationsTheoretical investigations



Excluding thermal bath variables from microscopic dynamics Excluding thermal bath variables from microscopic dynamics

(Kubo approach, (Kubo approach, Rep. Progr. Phys. 1966, V.29, P.255 Rep. Progr. Phys. 1966, V.29, P.255))

j

jjj

j

j xqm

m

p

m

pH

222

2222

xqmp

mpq

xqmp

mpx

jjjj

jjj

jjjj

2

2

/

)(

/

Equations of Hamiltonian mechanicsEquations of Hamiltonian mechanics

2. Different methods to reconstruct stochastic 2. Different methods to reconstruct stochastic macrodynamicsmacrodynamics

xqq

xqmxm

jjjj

jjjj

22

2



0,22

,22

0000002

kjB

jkj

kjBjk

kjjjpq

Tk

m

ppTkqqm

After solving the second equation and substituting in the first one we findAfter solving the second equation and substituting in the first one we find

jjjjjjjj

t

jjjj tptqmdxtmxm sincoscos 0022

Because ofBecause of

we immediately obtain GLE and fluctuation-dissipation theoremwe immediately obtain GLE and fluctuation-dissipation theorem

Phenomenological approachPhenomenological approach

Basic principles of statistical mechanics:Basic principles of statistical mechanics: Equilibrium Gibbsian Equilibrium Gibbsian distributiondistribution

Microscopic time reversibilityMicroscopic time reversibility



dztvPtm

zvP

z

ztvPv

vmt

tvP

,,,

1,2

dz

m

zqPqP

z

zdq

vP

mv stst

v

st

0 2

We will try to describe the particle of mass We will try to describe the particle of mass mm interacting with non- interacting with non-Gaussian white thermal bath Gaussian white thermal bath (t)(t) of the temperature of the temperature T T by the Langevin by the Langevin

equation containing additive noise sourceequation containing additive noise source

tvvm )( where (v) is unknown nonlinear dissipation

We use the general Kolmogorov’s equation obtained in the paperWe use the general Kolmogorov’s equation obtained in the paperA. Dubkov and B. Spagnolo, Fluct. Noise Lett. 2005. V.5, P.L267A. Dubkov and B. Spagnolo, Fluct. Noise Lett. 2005. V.5, P.L267

Taking into account the evident condition Taking into account the evident condition (0) =0=0 we find in asymptoticswe find in asymptotics



wherewhere

Tk

mvcvP

Bst 2

exp2

0

If the moments of the kernel function are finiteIf the moments of the kernel function are finite

we arrive atwe arrive at

dzzzB n

n 22

where where

1122/1

22

!2n Bnn

B

n vTk

mH

nTmk

Bv

is the equilibrium Maxwell is the equilibrium Maxwell distributiondistribution

are Hermitian polynomials are Hermitian polynomials 2/2/ 22

1)( zn

nzn

n edz

dezH



i

ii ttat

For white Gaussian noise sourceFor white Gaussian noise source (z(z))==2D2D((zz))

we havewe have

For Poissonian white noiseFor Poissonian white noise

with Gaussian distribution of amplitudeswith Gaussian distribution of amplitudes

2

2

2exp

2

1)(

x

xPa

TkDvv B/,

verfverfe

mv v

12

2

zerfTmkTkm BB ,/,2/ 2

is the error functionis the error function



In accordance with Stratonovich’s theoryIn accordance with Stratonovich’s theoryR.L. Stratonovich, Nonlinear Nonequilibrium R.L. Stratonovich, Nonlinear Nonequilibrium Thermodynamics. Springer-Verlag, Berlin, 1992Thermodynamics. Springer-Verlag, Berlin, 1992

3. Additive of multiplicative?3. Additive of multiplicative?

the stochastic Langevin equation should be multiplicative!the stochastic Langevin equation should be multiplicative!

tvvvm )(

From Kolmogorov’s equation we find for such a caseFrom Kolmogorov’s equation we find for such a case

tvdzPvv

zvv

zz

ztvPv

vmt

tvP,1exp,

1,2



If we put If we put P/ P/ t=0 t=0 we obtain complex relation between three functionswe obtain complex relation between three functions

is the Maxwell equilibrium distributionis the Maxwell equilibrium distribution vPst

01exp2

vdzPv

dv

dzv

dv

dz

z

zmvPv

dv

dstst

wherewhere

Choosing the statistics of noise Choosing the statistics of noise ((zz) ) we have the relationship between we have the relationship between the nonlinear dissipation and velocity-dependent noise intensitythe nonlinear dissipation and velocity-dependent noise intensity

For Poissonian noise we arrive atFor Poissonian noise we arrive at

01exp

vPdzv

dv

dav

dv

damvPv

dv

dst

ast



ConclusionsConclusions

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 2. The construction of physically correcting stochastic Langevin equation corresponding to the non-Gaussian Langevin equation corresponding to the non-Gaussian thermal bath and the nonlinear friction requests introducing a thermal bath and the nonlinear friction requests introducing a multiplicative noise sourcemultiplicative noise source..

3. This noise source should be non-Gaussian3. This noise source should be non-Gaussian..

4. Using the Gibbsian form of the equilibrium distribution one 4. Using the Gibbsian form of the equilibrium distribution one can find only relation between the nonlinear friction and the can find only relation between the nonlinear friction and the velocity-dependent noise intensityvelocity-dependent noise intensity..

5. Solution of this unsolved problem in the noise theory opens a way to the new realm of Brownian motion.

Documents

The Problem of Constructing Phenomenological Equations for Subsystem Interacting with non-Gaussian Thermal Bath Alexander Dubkov Nizhniy Novgorod State