Thermalization of Random Motion in Weakly Confining Potentials

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 < Tmax = 220/kB , where 20 sets an energy scale. For T ≥ Tmax no equilibrium pdf exists. We depart from a folk lore assumption that a thermodynamical system has a state characterized by a probability density [1]. That amounts to studying (random) dynamical systems in terms of time-dependent probability density functions (pdfs) and discovering whether and how the system may approach a state of thermodynamical equilibrium. Localization properties of pdfs, both far from and at equilibrium, can be quantified in terms of Shannon entropy. Admissible dynamical equilibria can be inferred from various variational principles. At this point we invoke a classification of maximum entropy principles (MEP) as given in [2]. Let us look for pdfs that derive from so-called first inverse MEP: given a pdf ρ(x), choose an appropriate set of constraints such that ρ(x) is obtained if Shannon measure of entropy is maximized (strictly speaking, extremized) subject to those constraints. Namely if a system evolves in a potential V (x) (at the moment, we consider a coordinate x to be dimensionless), we can introduce the following 288 P. Garbaczewski and V. Stephanovich