Lévy flights in a steep potential well displaying non-Gibbs-Boltzmann statistics

Abstract

We study analytically and numerically the probability density function in the stationary state of non-linear oscillators which are subjected to Lévy white noise and confined by a steep symmetric potential. The probability density function transforms from unimodality to bimodality or from bimodality to trimodality when the potential transforms from single well to double well; especially, the probability density function shows a peak at the saddle point of the potential. This result is far from the Gibbs-Boltzmann statistics.