Alexander Dubkov

1. UPoN 2008 Lyon (France),June2-6 The Problem of Constructing Phenomenological Equations for Subsystem Interacting with non-Gaussian Thermal Bath Alexander Dubkov Nizhniy Novgorod State University, Russia Peter Hänggi and Igor Goychuk Institut für Physik, Universität Augsburg, Germany This work was supported by RFBR grant08-02-01259

2. 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? Stratonovich’s approach to constructing Fokker-Planck equations • Conclusions UPoN 2008Lyon (France),June 2-62

3. The Problem of Constructing Phenomenological Equations for SubsystemInteracting with non-Gaussian Thermal Bath 1. Introduction The main problem of phenomenological theory: construction of the thermodynamicallycorrect stochastic equations for variables of subsystem interacting with thermal bath CENTRAL LIMIT THEOREMGAUSSIAN THERMAL BATH Classic Langevin equation with white Gaussian random force Einstein’s relation UPoN 2008Lyon (France),June 2-63

4. 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 • Newton’s nonlinear friction ((v)~|v|) UPoN 2008Lyon (France),June 2-64

5. 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 UPoN 2008Lyon (France),June 2-65

6. 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  UPoN 2008Lyon (France),June 2-66

7. 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-dissipation theorem Phenomenological approach Equilibrium Gibbsian distribution  Basic principles of statistical mechanics:  Microscopic time reversibility UPoN 2008Lyon (France),June 2-67

8. 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 Kolmogorov’s 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)=0we find in asymptotics UPoN 2008Lyon (France),June 2-68

9. 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 UPoN 2008Lyon (France),June 2-69

10. 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 UPoN 2008Lyon (France),June 2-610

11. The Problem of Constructing Phenomenological Equations for SubsystemInteracting with non-Gaussian Thermal Bath 3. Additive of multiplicative? In accordance with Stratonovich’s theory R.L. Stratonovich, Nonlinear Nonequilibrium Thermodynamics. Springer-Verlag, Berlin, 1992 the stochastic Langevin equation should be multiplicative! From Kolmogorov’s equation we find for such a case UPoN 2008Lyon (France),June 2-611

12. The Problem of Constructing Phenomenological Equations for SubsystemInteracting with non-Gaussian Thermal Bath If we put P/ t=0 we obtain complex relation between three functions where is the Maxwell equilibrium distribution 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 UPoN 2008Lyon (France),June 2-612

13. 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. UPoN 2008Lyon (France),June 2-613