Spectral densities from asymptotic solutions of the Fokker-Planck equation

By considering Fokker-Planck equation in the asymptotic limit, that is when strong potential act on the particles, a simplified description of its solutions is achievable. The asymptotic behaviour of the Fokker-Planck equation is analyzed for one-dimensional motions in the context of the correlation functions formalism. Much emphasis is placed on the single-minimum potential and analytical relations are derived in all the range of friction, from the zero-friction limit where the process of velocity relaxation is absent, to the diffusional regime. The planar rotator is used as a test case for comparing the asymptotic form of the spectral densities with the numerical solution of the Fokker-Planck operator. By incorporating known results of the theory of activated rate processes, the analysis is extended to multiminima potential