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 be also 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 \it only \rm in a bounded temperature interval $0\leq T < T_{max} =2\epsilon_0/k_B$, where $\epsilon_0$ sets an energy scale. For $T \geq T_{max}$ no equilibrium pdf exists.