The Problem of Constructing Phenomenological Equations for Subsystem Interacting with non-Gaussian Thermal Bath Alexander Dubkov Nizhniy Novgorod State University, Russia UPoN 2008 Lyon (France), June 2-6 This work was supported by RFBR grant Peter Hänggi and Igor Goychuk Institut für Physik, Universität Augsburg, Germany

OUTLINE IntroductionIntroduction Different methods to obtain a stochastic Langevin equations with Gaussian thermal bathDifferent methods to obtain a stochastic Langevin equations with Gaussian thermal bath Constructing the Langevin equation for Brownian particle interacting with non-Gaussian thermal bathConstructing the Langevin equation for Brownian particle interacting with non-Gaussian thermal bath Additive or multiplicative noise? Stratonovich’s approach to constructing Fokker-Planck equationsAdditive or multiplicative noise? Stratonovich’s approach to constructing Fokker-Planck equations ConclusionsConclusions The Problem of Constructing Phenomenological Equations for Subsystem Interacting with non-Gaussian Thermal Bath UPoN 2008Lyon (France),June 2-62 UPoN 2008 Lyon (France), June 2-62

The Problem of Constructing Phenomenological Equations for Subsystem Interacting with non-Gaussian Thermal Bath UPoN 2008Lyon (France),June 2-63 UPoN 2008 Lyon (France), June 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 Einstein’s relation Classic Langevin equation with white Gaussian random force

The Problem of Constructing Phenomenological Equations for Subsystem Interacting with non-Gaussian Thermal Bath UPoN 2008Lyon (France),June 2-64 UPoN 2008 Lyon (France), June 2-64 Why non-Gaussian thermal bath? Particle collisions with molecules of solvent (Poissonian noise)Particle collisions with molecules of solvent (Poissonian 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 molecular vibrations in molecular solidsAnharmonic molecular vibrations in molecular solids Newton’s nonlinear friction (  (v)~|v|)Newton’s nonlinear friction (  (v)~|v|) Generalized Langevin equation (GLE) of Kubo-Mori type Fluctuation-dissipation theorem:

The Problem of Constructing Phenomenological Equations for Subsystem Interacting with non-Gaussian Thermal Bath UPoN 2008Lyon (France),June 2-65 UPoN 2008 Lyon (France), June 2-65 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 V.29. P.888 A knowledge of nonlinear dissipative flow is not sufficient to reconstruct the original stochastic dynamics P. H ä nggi, Phys. Rev. A V.25. P.1130 Nonlinear stochastic models of oscillator systems G.N. Bochkov and Yu.E. Kuzovlev, Radiophys. and Quant. Electron V.21. P.1019 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 V.36. P.42; Phys. Rev. A V.13. P.556 Theoretical investigations

The Problem of Constructing Phenomenological Equations for Subsystem Interacting with non-Gaussian Thermal Bath UPoN 2008Lyon (France),June 2-66 UPoN 2008 Lyon (France), June 2-66 Excluding thermal bath variables from microscopic dynamics (Kubo approach, Rep. Progr. Phys. 1966, V.29, P.255 ) Equations of Hamiltonian mechanics 2. Different methods to reconstruct stochastic macrodynamics 

The Problem of Constructing Phenomenological Equations for Subsystem Interacting with non-Gaussian Thermal Bath UPoN 2008Lyon (France),June 2-67 UPoN 2008 Lyon (France), June 2-67 After solving the second equation and substituting in the first one we find Because of we immediately obtain GLE and fluctuation-dissipation theorem Phenomenological approach Basic principles of statistical mechanics:  Equilibrium Gibbsian distribution  Microscopic time reversibility

The Problem of Constructing Phenomenological Equations for Subsystem Interacting with non-Gaussian Thermal Bath UPoN 2008Lyon (France),June 2-68 UPoN 2008 Lyon (France), June 2-68 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 Kolmogorov’s equation obtained in the paper A. Dubkov and B. Spagnolo, Fluct. Noise Lett V.5, P.L267 Taking into account the evident condition =0 we find in asymptotics Taking into account the evident condition  (0) =0 we find in asymptotics

The Problem of Constructing Phenomenological Equations for Subsystem Interacting with non-Gaussian Thermal Bath UPoN 2008Lyon (France),June 2-69 UPoN 2008 Lyon (France), June 2-69 where If the moments of the kernel function are finite we arrive at where is the equilibrium Maxwell distribution are Hermitian polynomials

The Problem of Constructing Phenomenological Equations for Subsystem Interacting with non-Gaussian Thermal Bath UPoN 2008Lyon (France),June UPoN 2008 Lyon (France), June For white Gaussian noise source  (z)=2D  (z) we have For Poissonian white noise with Gaussian distribution of amplitudes is the error function

The Problem of Constructing Phenomenological Equations for Subsystem Interacting with non-Gaussian Thermal Bath UPoN 2008Lyon (France),June UPoN 2008 Lyon (France), June In accordance with Stratonovich’s theory R.L. Stratonovich, Nonlinear Nonequilibrium Thermodynamics. Springer-Verlag, Berlin, Additive of multiplicative? the stochastic Langevin equation should be multiplicative! From Kolmogorov’s equation we find for such a case

The Problem of Constructing Phenomenological Equations for Subsystem Interacting with non-Gaussian Thermal Bath UPoN 2008Lyon (France),June UPoN 2008 Lyon (France), June If we put  P/  t=0 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

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