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Lévy flights in a steep potential well. (English) Zbl 1157.82305

Summary: Lévy flights in steeper than harmonic potentials have been shown to exhibit finite variance and a critical time at which a bifurcation from an initial monomodal to a terminal bimodal distribution occurs. In this paper, we present a detailed study of Lévy flights in potentials of the type \(U(x) \sim |x|c\) with \(c>2\). Apart from the bifurcation into bimodality, we find the interesting result that for \(c>4\) a trimodal transient exists due to the temporal overlap between the decay of the central peak around the initial \(\delta\)-condition and the building up of the two emerging side-peaks, which are characteristic for the stationary state. Thus, for certain system parameters there exists a transient trimodal distribution of the Lévy flight. These properties of Lévy flights in external potentials of the power-law type can be represented by certain phase diagrams. We also present details about the proof of multimodality and the numerical procedures to establish the probability distribution of the process.

MSC:

82B05 Classical equilibrium statistical mechanics (general)
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