Integrability of Hamiltonians associated with Fokker-Planck equations

Publisher Website
Google Scholar
Abstract
Stochastic dynamical models described by Fokker-Planck equations, in the limit of weak noise, can be formally associated with Hamiltonian dynamical systems. Of special interest are ‘‘Fokker-Planck Hamiltonians’’ with a certain smooth separatrix at zero energy, since a differentiable macroscopic potential was shown to exist in this case. In the present paper, integrability of Fokker-Planck Hamiltonians with two degrees of freedom is investigated, with the aim of identifying cases in which a smooth potential exists and cases in which the eigenvalue problem of the Fokker-Planck operator becomes separable. Additional first integrals of polynomial and nonpolynomial form and also explicitly time-dependent conserved quantities are obtained. The singular-point analysis testing for the Painlevé or the weak Painlevé property and polynomial conserved quantities in dynamical systems is applied. This method is found to be useful for the purpose of identifying solvable special cases in classes of Fokker-Planck models with free parameters. However, the utility of this method in a search for systems with a smooth potential is found to be severely limited because it turns out that complete integrability with polynomial conserved quantities is a much stronger requirement than the existence of a smooth separatrix at zero energy.