Variable separation without preselection and non-spectral relaxation in Fokker-Planck and generalized Fokker-Planck equations

The relaxation of a dissipative system to its equilibrium state often shows a multiexponential pattern with relaxation rates, which are typically considered to be independent of the initial conditions. The rates typically follow from the spectrum of a Hermitian operator obtained by a similarity transformation of the initial Fokker-Planck operator. However, some initial conditions are mapped by this similarity transformation to functions which grow at infinity. These cannot be expanded in terms of the eigenfunctions of a Hermitian operator, and show different relaxation patterns (non-spectral relaxation). Considering the exactly solvable examples of Gaussian and generalized Lévy Ornstein-Uhlenbeck processes (OUPs) we have shown that the relaxation rates belong to the Hermitian spectrum only if the initial condition belongs to the domain of attraction of the stable distribution defining the noise. In this project we are going to revisit the analytically solvable Ornstein-Uhlenbeck problem and discuss in detail the properties of the similarity transformation in the real space. Moreover, we plan to discuss in more detail the nature of selection rules for spectral and non-spectral relaxation in this case, consider a wider class of problems for which the exact solution is still possible and look for the best practical re-expansion procedures giving non-spectral rates in such cases, and investigate general cases in which the exact solutions are not known using the concept of regular variation.

Principal Investigators
Sokolov, Igor Prof. Dr. rer. nat. (Details) (Theoretical Physics / Statistical Physics and Nonlinear Dynamics)

DFG: Sachbeihilfe

Duration of Project
Start date: 06/2015
End date: 05/2016

Last updated on 2020-20-03 at 23:20