Overshooting and undershooting subordination scenario for fractional two-power-law relaxation responses

Abstract

In this paper, we propose a transparent subordination approach to anomalous diffusion processes underlying the nonexponential relaxation. We investigate properties of a coupled continuous-time random walk that follows from modeling the occurrence of jumps with compound counting processes. As a result, two different diffusion processes corresponding to over- and undershooting operational times, respectively, have been found. We show that within the proposed framework, all empirical two-power-law relaxation patterns may be derived. This work is motivated by the so-called "less typical" relaxation behavior observed, e.g., for gallium-doped Cd0.99Mn0.01Te mixed crystals.

Weakly anomalous diffusion with non-Gaussian propagators

“…The results we report below show clearly how long-range memory effects can change H and the propagator independently. Our results also have a bearing on the family of autoregressive and heteroscedastic processes, some of which have a bearing on anomalous diffusion [8][9][10][11][12][13][14][15][16][17][18][19][20][21].…”

“…For α < 0 (p < 1/2), the escape regime disappears on the δ = 0 line, but the log-periodicity remains. For α > 0 (p > 1/2), the solution for x is given by (10). Again, δ = 0 divides the p > 1/2 region in two: one below (escape regime, δ > 0) for which x(t) → ∞ and one above (normally diffusive, δ < 0) for which x(t) → 0, asymptotically.…”

Clustered continuous-time random walks: diffusion and relaxation consequences

“…The names refer to the property that, for any time t ≥ 0, 8) so that the under-and over-shooting processes under-and over-estimate the exact instant of time t ≥ 0, respectively. As shown in Weron et al (2010) and Jurlewicz et al (2011), both processes are 1-self-similar and, for a fixed t ≥ 0, the distribution of H − g (t) is the same as tJ for the random variable J with the probability density function (PDF)…”

“…The relaxation patterns f(t) are determined by the stochastic properties of the jumps, the inter-jump times and the detailed construction of the counting process. For the cluster CTRW and OCTRW models, the anomalous diffusion fronts W ± (t) in (2.6) and (2.7) have frequency-domain dielectric susceptibility functions that can be identified as the JWS and HN functions, equations (1.3) and (1.2), respectively, see and Weron et al (2010). The characteristic material constant u p , appearing in both functions, takes the form…”

“…However, the derivation of the HN function (1.2), based on the fractional Fokker-Planck equation (Kalmykov et al 2004) and the CTRW (Weron et al 2005;Jurlewicz et al 2008), does not lead to values of g > 1. Only recently, by employing the subordination approach Weron et al 2010), developed in the last decade in the theory of anomalous diffusion (Sokolov 2001(Sokolov , 2002Meerschaert et al 2002;Meerschaert & Scheffler 2004;Piryatinska et al 2005;Magdziarz & Weron 2006;Sokolov & Klafter 2006;Magdziarz et al 2007Magdziarz et al , 2008Lubelski et al 2008), has the effective picture underlying all typical and less typical relaxation patterns been found. In contrast to earlier studies (Weron & Kotulski 1996;Metzler et al 1999) where the anomalous-diffusion approach (based on a decoupled CTRW) has led to CC relaxation only, the compound subordination framework allows one to derive rigorously Weron et al 2010) the original HN function as well as the novel version…”