Thermalization of Random Motion in Weakly Confining Potentials

Abstract

We show that in weakly confining conservative force fields, a subclass of diffusion-type (Smoluchowski) processes, admits a family of "heavy-tailed" non-Gaussian equilibrium probability density functions (pdfs), with none or a finite number of moments. These pdfs, in the standard Gibbs-Boltzmann form, can also be inferred directly from an extremum principle, set for Shannon entropy under a constraint that the mean value of the force potential has been a priori prescribed. That enforces the corresponding Lagrange multiplier to play the role of inverse temperature. Weak confining properties of the potentials are manifested in a thermodynamical peculiarity that thermal equilibria can be approached only in a bounded temperature interval 0 ≤ T < T max = 2ϵ0/kB, where ϵ0 sets an energy scale. For T ≥ T max no equilibrium pdf exists.