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Applications of statistical mechanics to nonbrownian random motion 
Ryszard Kutner , Krzysztof Wysocki 
Faculty of Physics, University of Warsaw (FPUW), Pasteura 5, Warsaw 02093, Poland 
Abstract 
We analysed discrete and continuous Weierstrass–Mandelbrot representations of the Lévy flights occasionally interrupted by spatial localizations. We chose the discrete representation to easily detect by Monte Carlo simulation which stochastic quantity could be a candidate for describing the real processes. We found that the particle propagator is able to reveal surprisingly close, stable longrange algebraic tail. Unfortunately, long flights present in the system make, in practice, the particle meansquare displacement an irregular steplike function; such a behavior was expected since it is an experimental reminiscence of divergence of the meansquare displacement, predicted by the theory. We developed the continuous representation in the context of random motion of a particle in an amorphous environment; we established a correspondence between the stochastic quantities of both representations in which the latter quantities contain some material constants. The material constants appear due to the thermal average of the spacedependent stretch exponent which defines the probability of the particle passing a given distance. This averaging was performed for intermediate or even high temperatures, as well as for low or even intermediate internal friction regimes where long but not extremely long flights are readily able to construct a significant part of the Lévy distribution. This supplies a kind of selfcutoff of the length of flights. By way of example, we considered a possibility of observing the Lévy flights of hydrogen in amorphous lowconcentration, hightemperature Pd_{85}Si_{15}H_{7.5} phase; this conclusion is based on the results of a real experiment (Driesen et al., in: Janot et al. (Eds.), Atomic Transport and Defects in Metals by Neutron Scattering, Proceedings in Physics, Vol. 10, Springer, Berlin, 1986, p. 126; Richter et al., Phys. Rev. Lett. 57 (1986) 731; Driesen, Doctoral Thesis, Antwerpen University, 1987), performed by detecting the incoherent quasielastic scattering of thermal neutrons. We emphasize that the observed HWHM ∼k^{
β
}, where exponent β is distinctly smaller than 2, could be caused by these long flights of hydrogen.
Physica A 274 (1999), 67. 
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Presentation: Article at Econophysics group of Ryszard Kutner, by Ryszard Kutner Submitted: 20160329 14:18 Revised: 20160329 14:18 