Cusp of Non-Gaussian Density of Particles for a Diffusing Diffusivity Model

Abstract

We study a two state “jumping diffusivity” model for a Brownian process alternating between two different diffusion constants, D+>D−, with random waiting times in both states whose distribution is rather general. In the limit of long measurement times, Gaussian behavior with an effective diffusion coefficient is recovered. We show that, for equilibrium initial conditions and when the limit of the diffusion coefficient D−⟶0 is taken, the short time behavior leads to a cusp, namely a non-analytical behavior, in the distribution of the displacements P(x,t) for x⟶0. Visually this cusp, or tent-like shape, resembles similar behavior found in many experiments of diffusing particles in disordered environments, such as glassy systems and intracellular media. This general result depends only on the existence of finite mean values of the waiting times at the different states of the model. Gaussian statistics in the long time limit is achieved due to ergodicity and convergence of the distribution of the temporal occupation fraction in state D+ to a δ-function. The short time behavior of the same quantity converges to a uniform distribution, which leads to the non-analyticity in P(x,t). We demonstrate how super-statistical framework is a zeroth order short time expansion of P(x,t), in the number of transitions, that does not yield the cusp like shape. The latter, considered as the key feature of experiments in the field, is found with the first correction in perturbation theory.

Ergodic property of random diffusivity system with trapping events

“…Therefore, the form does not depend on the initial condition while the latter does [73,74]. However, if the system starts from equilibrium initial conditions, the EAMSD and TAMSD would behave similarly with respect to lag time [63,[75][76][77]. More connections between the random diffusivity model and other anomalous diffusion processes will be discussed in the future.…”

Novel anomalous diffusion phenomena of underdamped Langevin equation with random parameters

“…The time average is in some sense an equilibrium measure, while the ensemble average is not. The different diffusion behavior resulting from the discrepant initial ensemble has been observed in the context of single file diffusion [63], diffusing diffusivity model [64], Lévy walk [65,66,67], biased random walk [45,68]. For the underdamped Langevin equation with random relaxation timescale τ , we also study the effects of different initial velocity v 0 , which is also the main part of this paper.…”

“…distribution at t = 0, then it is named equilibrated initial ensemble [74]. Different initial ensemble usually leads to different diffusion behavior, which has been discussed a lot [45,63,64,65,66,67,68]. It is noteworthy that the way of initial ensemble affecting the diffusion behavior is quite special for the underdamped Langevin equation (3) with random relaxation timescale τ .…”

Random diffusivity scenarios behind anomalous non-Gaussian diffusion

“…Then, the porous media problem is a particular case where γ is coupled to ν. The PDF distribution of tracers (23) has numerous applications in complex systems, for example in modeling of protein diffusion within bacteria [29], parliamentary presence data [77] and stock markets [78].…”

“…Its impact can be addressed through the superstatistical approach proposed by C. Beck and E. G. D. Cohen to extend statistical mechanics to complex heterogeneous environments. Superstatistics has been applied to run-and-tumble particles [13], animal movement [14][15][16], metapopulation extinction dynamics [17], time series analyses [18][19][20][21], and many other cases [22][23][24][25][26][27][28]. The superstatistics of fBm has been recently developed, providing theoretical support for various experimental ob-servations, such as, protein diffusion in bacteria [29,30], micro-particles in a bi-dimensional system with disordered distribution of pillars [31], and tracer diffusion in mucin hydrogels [32] (see Refs.…”