0 Non - Boltzmann Equilibrium Probability Densities for Non - Linear Lévy Oscillator

We study, both analytically and by numerical modeling the equilibrium probability density function for an non-linear Lévy oscillator with the Lévy index α, 1 ≤ α ≤ 2, and the potential energy x 4. In particular, we show that the equilibrium PDF is bimodal and has power law asymptotics with the exponent −(α + 3). 1 Starting equations Recently, kinetic equations with fractional derivatives have attracted attention as a possible tool for the description of anomalous diffusion and relaxation phenomena, see, e.g., recent review [1], semi-review papers [2],[3],[4] and references on earlier studies therein. It was also recognized [5], [6] that the fractional kinetic equations may be viewed as " hydrodynamic " (that is, long-time and long-space) limits of the CTRW (Continuous Time Random Walk) scheme [7], which was successfully applied to the description of anomalous diffusion phenomena in many areas, e.g., turbulence [8], disordered medium [9], intermittent chaotic systems [10], etc. However, the kinetic equations have two advantages over a random walk approach: firstly, they allows one to explore various boundary conditions (e.g., reflecting and/or absorbing) and, secondly, to study diffusion and/or relaxation phenomena in external fields, both possibilities are difficult to realize in the framework of CTRW (we point, however, to the paper [11], in which a fractional kinetic equation was obtained from generalized CTRW). Fractional kinetic equations can be divided into three classes: the first one, describing Markovian processes, contains equations with fractional space or velocity derivatives and the first time derivative, the second one, describing non-Markovian processes, contains equations with fractional time derivative, and the third class, naturally, contains both fractional space and time derivatives, as well. In this paper we deal with a one-dimensional kinetic equation belonging to the first class, namely, with the Fractional Symmetric Einstein-Smoluchowski Equation (FSESE), which, from one hand, is a natural generalization of the diffusion-like equation with the symmetric fractional space derivative [3],[12] and, from the other hand, is a Markovian generalization of the Einstein-Smoluchowski kinetic equation, which describes a motion of a particle subjected to a white Gaussian noise in a strong friction limit, see, e.g., [13]. From this point of view, the FSESE describes a motion of a particle subjected to a white Lévy noise, also in a strong friction limit [14]. In dimensionless units the one-dimensional FSESE has the form ∂f ∂t = − ∂ ∂x (F f) + ∂ α f ∂ |x| α , t …