Stationary Solutions of the Fractional Kinetic Equation with a Symmetric Power-Law Potential

The properties of stationary solutions of the one-dimensional fractional Einstein–Smoluchowski equation with a potential of the form $x^{2m+2}$, $m=1,2,\dots$, and of the Riesz spatial fractional derivative of order $\alpha$, $1\leq\alpha\leq2$ are studied analytically and numerically. We show that for $1\leq\alpha<2$, the stationary distribution functions have power-law asymptotic approximations decreasing as $x^{-(\alpha+2m+1)}$ for large values of the argument. We also show that these distributions are bimodal.