Fractional Master Equation Research Articles

Anomalous transport in a one-dimensional Lorentz gas model

Employing the generalized master equation proposed in [R. Friedrich et al., Phys. Rev. Lett. 96, 230601 (2006)], we derive a kinetic equation for a random kick model. For a particular choice of the time evolution kernel, a fractional master equation is obtained, which can be related to a Levy walk. In one dimension, we use this model to describe a stochastic Lorentz gas with an annealed disorder. Exact moment relations are obtained in Laplace space, and the long-time behavior of the moments is discussed. The results are compared to those of related models.

Fractional master equation: non-standard analysis and Liouville–Riemann derivative

Fractional master equations may be defined either by means of Liouville–Riemann (L–R) fractional derivative or via non-standard analysis. The first approach describes processes with long-range dependence whilst the second approach deals with processes involving independent increments. The present papers put in evidence some of the differences between these two modellings, and to this end it especially considers more fractional Poisson processes.

Fractional Integral Representation of Master Equation

Using the definition of Liouville–Riemann (L–R) fractional integral operator, master equation can be represented in the domain of fractal time evolution with a critical exponent a (0<a⩽1) . The relation between the continuous time random walks (CTRW) and fractional master equation (FME) has been achieved by obtaining the corresponding waiting time density (WTD) ψ (t) . The latter is obtained in a closed form in terms of the generalized Mittag–Leffler (M–L) function. The asymptotic expansion of the (M–L) function show the same behavior considered in the theory of random walk. Applying the Fourier and Laplace–Mellin transforms to (FME) , one obtains the solution, in closed form, in terms of the Fox function.

EXACT SOLUTIONS FOR A CLASS OF FRACTAL TIME RANDOM WALKS

Fractal time random walks with generalized Mittag-Leffler functions as waiting time densities are studied. This class of fractal time processes is characterized by a dynamical critical exponent 0&lt;ω≤1, and is equivalently described by a fractional master equation with time derivative of noninteger order ω. Exact Greens functions corresponding to fractional diffusion are obtained using Mellin transform techniques. The Greens functions are expressible in terms of general H-functions. For ω&lt;1 they are singular at the origin and exhibit a stretched Gaussian form at infinity. Changing the order ω interpolates smoothly between ordinary diffusion ω=1 and completely localized behavior in the ω→0 limit.

Fractional master equations and fractal time random walks.

Fractional master equations containing fractional time derivatives of order 0\ensuremath{\le}1 are introduced on the basis of a recent classification of time generators in ergodic theory. It is shown that fractional master equations are contained as a special case within the traditional theory of continuous time random walks. The corresponding waiting time density \ensuremath{\psi}(t) is obtained exactly as \ensuremath{\psi}(t)=(${\mathit{t}}^{\mathrm{\ensuremath{\omega}}\mathrm{\ensuremath{-}}1}$/C)${\mathit{E}}_{\mathrm{\ensuremath{\omega}},\mathrm{\ensuremath{\omega}}}$(-${\mathit{t}}^{\mathrm{\ensuremath{\omega}}}$/C), where ${\mathit{E}}_{\mathrm{\ensuremath{\omega}},\mathrm{\ensuremath{\omega}}}$(x) is the generalized Mittag-Leffler function. This waiting time distribution is singular both in the long time as well as in the short time limit.