Superdiffusive Klein-Kramers equation: Normal and ano malous time evolution and Lévy walk moments

Abstract

We introduce a fractional Klein-Kramers equation which describes sub-ballistic superdiffusion in phase space in the presence of a space-dependent external force field. This equation defines the differential Lévy walk model whose solution is shown to be non-negative. In the velocity coordinate, the probability density relaxes in Mittag-Leffler fashion towards the Maxwell distribution whereas in the space coordinate, no stationary solution exists and the temporal evolution of moments exhibits a competition between Brownian and anomalous contributions. Classically, Brownian stochastic transport processes in the phase space spanned by velocity v and coordinate x are described by the deterministic Klein-Kramers equation (KKE). In the low and high friction limits, the KKE reduces to the Rayleigh equation which describes the relaxation of the velocity probability density function (pdf) towards the Maxwell distribution, and the FokkerPlanck-Smoluchowski equation controlling the temporal approach of the Gibbs-Boltzmann equilibrium, respectively [1, 2]. The KKE is therefore a fundamental equation in non-equilibrium systems dynamics.Brownian transport is characterised through the Gaussian pdf and the linear time dependence x 2 (t) = 2Kt of the mean squared displacement in the force-free diffusion limit, its universality being guaranteed by the central limit theorem [2]. In a broad variety of systems, however, it has been found that correlations in space or time give birth to anomalous transport whose pdf is non-Gaussian and/or whose mean squared displacement is non-linear in time [3]. These systems include charge carrier transport in amorphous semiconductors [4], tracer dispersion in convection rolls and rotating flows [5,6] The systems we are interested in fall into the broad class whose force-free diffusion behaviour is characterised through the power-law form x 2 (t) ∝ t κ , which separates into subdiffusion (0 < κ < 1) and superdiffusion (κ > 1). The continuous time random walk (CTRW) model has proved to be a well-suited framework which accounts for such anomalous diffusion for the entire spectrum of κ [14]. Especially in the sub-ballistic superdiffusive domain 1 < κ < 2, Lévy walks which couple long flight times with a time cost have been a successful tool, e.g., in fluid dynamics [6,15]. The space-time coupling of Lévy walks leads to finite moments and they are therefore fundamentally different from Lévy flights which exhibit a diverging variance [16].In the presence of external force fields, the CTRW approach is less flexible. It has been realised that fractional equations constitute a tailor-made framework to formulate the underlying dynamics equations in coordinate and phase space; see, for instance, [16,17,18,19,20] and references therein. In the subdiffusive domain 0 < κ < 1, a fractional KKE was derived from CTRW models [21] and from the Chapman-Kolmogorov equation [22]. A similar consistent generalisation to systems in the regime 1 < κ < 2 is still outstanding. In this note, we propose the fractional KKE

Anomalous Fluctuation Relations

“…These results suggest that, for obtaining a stochastic model, the force-free fractional Klein-Kramers equation Eq. (1.31) needs to be generalized by including an external force as discussed by Metzler and Sokolov [99],…”

Fluctuation relations for anomalous dynamics generated by time-fractional Fokker–Planck equations

“…FFPE can be derived from stochastic equations of motion either by CTRWs [12,16] or by subordinated Langevin dynamics [47]. Quite a variety of them have been studied in the literature, both from a purely theoretical point of view and with respect to applications to experiments: Prominent examples are fractional Klein-Kramers equations that were used to analyse biological cell migration data [48][49][50]. Another type was designed to model the dynamics of tracer particles in random environments [51].…”

Towards deterministic equations for Lévy walks: The fractional material derivative

“…A question of fundamental interest is therefore the formulation of LWs in terms of deterministic equations. Whereas previous approaches [13,14] in terms of fractional Klein-Kramers equations could reproduce lower order moments of an LW, they were hampered by the fact that they could not describe the full pdf. The main complication on this way is the fact, that the overall LW process cannot be immediately considered as subordinated to a Wiener one (or to a simple random walk); however, as we proceed to show, it is exactly the strong correlation of the temporal and the spatial aspects of LWs which makes it possible to provide a description based on a process subordinated to a simple two-state Markovian process.…”