Langevin Approach to Lévy Flights In Fixed Potentials: Exact Results for Stationary Probability Distributions (original) (raw)
Physical Review E, 1999
We consider Lévy flights subject to external force fields. This anomalous transport process is described by two approaches, a Langevin equation with Lévy noise and the corresponding generalized Fokker-Planck equation containing a fractional derivative in space. The cases of free flights, constant force and linear Hookean force are analyzed in detail, and we corroborate our findings with results from numerical simulations. We discuss the non-Gibbsian character of the stationary solution for the case of the Hookean force, i.e. the deviation from Boltzmann equilibrium for long times. The possible connection to Tsallis's q-statistics is studied. 05.40.+j,05.60.+w,02.50.Ey,05.70.Ln
2001
The Fokker-Planck equation has been very useful for studying dynamic behavior of stochastic differential equations driven by Gaussian noises. However, there are both theoretical and empirical reasons to consider similar equations driven by strongly non-Gaussian noises. In particular, they yield strongly non-Gaussian anomalous diffusion which seems to be relevant in different domains of Physics. In this paper, we therefore derive a fractional Fokker-Planck equation for the probability distribution of particles whose motion is governed by a nonlinear Langevintype equation, which is driven by a Lévy stable noise rather than a Gaussian. We obtain in fact a general result for a Markovian forcing. We also discuss the existence and uniqueness of the solution of the fractional Fokker-Planck equation.
Entropy, 2018
The numerical solutions to a non-linear Fractional Fokker–Planck (FFP) equation are studied estimating the generalized diffusion coefficients. The aim is to model anomalous diffusion using an FFP description with fractional velocity derivatives and Langevin dynamics where Lévy fluctuations are introduced to model the effect of non-local transport due to fractional diffusion in velocity space. Distribution functions are found using numerical means for varying degrees of fractionality of the stable Lévy distribution as solutions to the FFP equation. The statistical properties of the distribution functions are assessed by a generalized normalized expectation measure and entropy and modified transport coefficient. The transport coefficient significantly increases with decreasing fractality which is corroborated by analysis of experimental data.
Steady-state Lévy flights in a confined domain
Physical Review E, 2008
We derive the generalized Fokker-Planck equation associated with a Langevin equation driven by arbitrary additive white noise. We apply our result to study the distribution of symmetric and asymmetric Lévy flights in an infinitely deep potential well. The fractional Fokker-Planck equation for Lévy flights is derived and solved analytically in the steady state. It is shown that Lévy flights are distributed according to the beta distribution, whose probability density becomes singular at the boundaries of the well. The origin of the preferred concentration of flying objects near the boundaries in nonequilibrium systems is clarified.
Lévy anomalous diffusion and fractional Fokker–Planck equation
Physica A: Statistical Mechanics and its Applications, 2000
We demonstrate that the Fokker-Planck equation can be generalized into a 'Fractional Fokker-Planck' equation, i.e. an equation which includes fractional space differentiations, in order to encompass the wide class of anomalous diffusions due to a Lévy stable stochastic forcing. A precise determination of this equation is obtained by substituting a Lévy stable source to the classical gaussian one in the Langevin equation. This yields not only the anomalous diffusion coefficient, but a non trivial fractional operator which corresponds to the possible asymmetry of the Lévy stable source. Both of them cannot be obtained by scaling arguments. The (mono-) scaling behaviors of the Fractional Fokker-Planck equation and of its solutions are analysed and a generalization of the Einstein relation for the anomalous diffusion coefficient is obtained. This generalization yields a straightforward physical interpretation of the parameters of Lévy stable distributions. Furthermore, with the help of important examples, we show the applicability of the Fractional Fokker-Planck equation in physics.
Journal of Mathematical Physics, 2014
We study generalized fractional Langevin equations in the presence of a harmonic potential. General expressions for the mean velocity and particle displacement, the mean squared displacement, position and velocity correlation functions, as well as normalized displacement correlation function are derived. We report exact results for the cases of internal and external friction, that is, when the driving noise is either internal and thus the fluctuation-dissipation relation is fulfilled or when the noise is external. The asymptotic behavior of the generalized stochastic oscillator is investigated, and the case of high viscous damping (overdamped limit) is considered. Additional behaviors of the normalized displacement correlation functions different from those for the regular damped harmonic oscillator are observed. In addition, the cases of a constant external force and the force free case are obtained. The validity of the generalized Einstein relation for this process is discussed. The considered fractional generalized Langevin equation may be used to model anomalous diffusive processes including single file-type diffusion. C 2014 AIP Publishing LLC.
The fractional-order governing equation of Lévy Motion
Water Resources Research, 2000
A governing equation of stable random walks is developed in one dimension. This Fokker-Planck equation is similar to, and contains as a subset, the second-order advection dispersion equation (ADE) except that the order (␣) of the highest derivative is fractional (e.g., the 1.65th derivative). Fundamental solutions are Lévy's ␣-stable densities that resemble the Gaussian except that they spread proportional to time 1/␣ , have heavier tails, and incorporate any degree of skewness. The measured variance of a plume undergoing Lévy motion would grow faster than Fickian plume, at a rate of time 2/␣ , where 0 Ͻ ␣ Յ 2. The equation is parsimonious since the parameters are not functions of time or distance. The scaling behavior of plumes that undergo Lévy motion is accounted for by the fractional derivatives, which are appropriate measures of fractal functions. In real space the fractional derivatives are integrodifferential operators, so the fractional ADE describes a spatially nonlocal process that is ergodic and has analytic solutions for all time and space.
Can one see a competition between subdiffusion and Lévy flights? A case of geometric-stable noise
2008
Competition between subdiffusion and Lévy flights is conveniently described by the fractional Fokker-Planck equation with temporal and spatial fractional derivatives. The equivalent approach is based on the subordinated Langevin equation with stable noise. In this paper we examine the properties of such Langevin equation with the heavy-tailed noise belonging to the class of geometric stable distributions. In particular, we consider two physically relevant examples of geometric stable noises, namely Linnik and Mittag-Leffler. We describe in detail a numerical algorithm for visualization of subdiffusion coexisting with Lévy flights. Using Monte Carlo simulations we demonstrate the realizations as well as the probability density functions of the considered anomalous diffusion process.