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Lévy flights in confining environments: random paths and their statistics. (English) Zbl 1395.82089
Summary: We analyze a specific class of random systems that, while being driven by a symmetric Lévy stable noise, asymptotically set down at the Boltzmann-type equilibrium, represented by a probability density function (pdf) \(\rho_\ast(x) \sim \exp [- \Phi(x)]\). This behavior needs to be contrasted with the standard Langevin representation of Lévy jump-type processes. It is known that the choice of the drift function in the Newtonian form \(\sim - \nabla \Phi\) excludes the existence of the Boltzmannian pdf \(\sim \exp [- \Phi(x)]\) (Eliazar-Klafter no go theorem). In view of this incompatibility statement, our main goal here is to establish the appropriate path-wise description of the equilibrating jump-type process. A priori given inputs are (i) jump transition rates entering the master equation for \(\rho(x, t)\) and (ii) the target (invariant) pdf \(\rho_\ast(x)\) of that equation, in the Boltzmannian form. We resort to numerical methods and construct a suitable modification of the Gillespie algorithm, originally invented in the chemical kinetics context. The generated sample trajectories show up a qualitative typicality, e.g. they display structural features of jumping paths (predominance of small vs large jumps) specific to particular stability indices \(\mu \in(0, 2)\). The obtained random paths statistical data allow us to infer an associated pdf \(\rho(x, t)\) dynamics which stands for a validity check of our procedure. The considered exemplary Boltzmannian equilibria \(\sim \exp [- \Phi(x)]\) refer to (i) harmonic potential \(\Phi \sim x^2\), (ii) logarithmic potential \(\Phi \sim n \ln(1 + x^2)\) with \(n = 1, 2\) and (iii) locally periodic confining potential \(\Phi \sim \sin^2(2 \pi x), | x | \leq 2\), \(\Phi \sim(x^2 - 4), | x | > 2\).
MSC:
82B31 Stochastic methods applied to problems in equilibrium statistical mechanics
82B80 Numerical methods in equilibrium statistical mechanics (MSC2010)
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