Exact expression for the diffusion propagator in a family of time-dependent anharmonic potentials

TL;DR

This paper derives an exact analytical expression for the diffusion propagator in a specific family of time-dependent anharmonic potentials, providing insights into particle flow through barriers.

Contribution

It presents the first exact solution for the diffusion propagator in a class of time-dependent anharmonic potentials with a logarithmic term.

Findings

Exact expression for the diffusion propagator derived

Conditions for particle flow through the barrier analyzed

Analytical solutions obtained using Bessel functions

Abstract

We have obtained the exact expression of the diffusion propagator in the time-dependent anharmonic potential $V(x,t)={1/2}a(t)x^2+b\ln x$. The underlying Euclidean metric of the problem allows us to obtain analytical solutions for a whole family of the elastic parameter a(t), exploiting the relation between the path integral representation of the short time propagator and the modified Bessel functions. We have also analyzed the conditions for the appearance of a non-zero flow of particles through the infinite barrier located at the origin (b<0).