Sample-dependent first-passage-time distribution in a disordered medium

Trace of anomalous diffusion in a biased quenched trap model

Diffusion in a quenched heterogeneous environment in the presence of bias is considered analytically. The first-passage-time statistics can be applied to obtain the drift and the diffusion coefficient in periodic quenched environments. We show several transition points at which sample-to-sample fluctuations of the drifts or the diffusion coefficients remain large even when the system size becomes large, i.e., non-self-averaging. Moreover, we find that the disorder average of the diffusion coefficient diverges or becomes 0 when the corresponding annealed model generates superdiffusion or subdiffusion, respectively. This result implies that anomalous diffusion in an annealed model is traced by anomaly of the diffusion coefficients in the corresponding quenched model.

Fluctuations and self-averaging in random trapping transport: The diffusion coefficient

On the basis of a self-consistent cluster effective-medium approximation for random trapping transport, we study the problem of self-averaging of the diffusion coefficient in a nonstationary formulation. In the long-time domain, we investigate different cases that correspond to the increasing degree of disorder. In the regular and subregular cases the diffusion coefficient is found to be a self-averaging quantity-its relative fluctuations (relative standard deviation) decay in time in a power-law fashion. In the subdispersive case the diffusion coefficient is self-averaging in three dimensions (3D) and weakly self-averaging in two dimensions (2D) and one dimension (1D), when its relative fluctuations decay anomalously slowly logarithmically. In the dispersive case, the diffusion coefficient is self-averaging in 3D, weakly self-averaging in 2D, and non-self-averaging in 1D. When non-self-averaging, its fluctuations remain of the same order as, or larger than, its average value. In the irreversible case, the diffusion coefficient is non-self-averaging in any dimension. In general, with the decreasing dimension and/or increasing disorder, the self-averaging worsens and eventually disappears. In the cases of weak self-averaging and, especially, non-self-averaging, the reliable reproducible experimental measurements are highly problematic. In all the cases under consideration, asymptotics with prefactors are obtained beyond the scaling laws. Transition between all cases is analyzed as the disorder increases.

Non-self-averaging in random trapping transport: The diffusion coefficient in the fluctuation regime

The nonstationary diffusion of particles in a medium with static random traps or sinks is considered. The question of the self-averaging of the diffusion coefficient (or, equivalently, of the mean-square displacement) is addressed for the fluctuation regime in the long-time limit. The property of self-averaging is needed for the result of a single measurement to be representative and reproducible. It is demonstrated that the diffusion coefficient of the surviving particles is a strongly non-self-averaging quantity: In a d-dimensional system its reciprocal standard deviation grows with time exponentially ≈exp[const_{d,1}t^{d/(d+2)}]. The same result is reproduced in the "normalized" formulation "per one survivor on average." The case when all the particles, both the survivors and the trapped ones, are contributing to the diffusion coefficient and its variance is considered also. Non-self-averaging is demonstrated for this case as well, the fluctuations of the diffusion coefficient being of the same order as its average value. The critical dimension, above which the mean-field result becomes exact, is infinite-due to the drastic difference between the classes of trajectories, upon which the corresponding results are being built. In high dimensions the strong non-self-averaging of survivors is preserved. For the case of all the particles taken into account, the nonstrong non-self-averaging is retained for any finite dimension. However, for d→∞ the limiting value of the reciprocal standard deviation, calculated for all the particles, decreases to zero. This signifies restoration of the self-averaging in some sense. In all the cases, the time evolution of the average characteristics and of their variances is governed by the decaying concentration of the survivors in fluctuational cavities.

Quenched trap model on the extreme landscape: The rise of subdiffusion and non-Gaussian diffusion

Non-Gaussian diffusion has been intensively studied in recent years, which reflects the dynamic heterogeneity in the disordered media. The recent study on the non-Gaussian diffusion in a static disordered landscape suggests novel phenomena due to the quenched disorder. In this paper, we further investigate the random walk on this landscape under various effective temperatures μ, which continuously modulate the dynamic heterogeneity. We show in the long-time limit, the trap dynamics on the landscape is equivalent to the quenched trap model in which subdiffusion appears for μ<1. The non-Gaussian distribution of displacement has been analytically estimated for short t of which the stretched exponential tail is expected for μ≠1. Due to the localization in the ensemble of trajectory segments, an additional peak arises in P(x,t) around x=0 even for μ>1. Evolving in different timescales, the peak and the tail of P(x,t) are well split for a wide range of t. This theoretical paper reveals the connections among the subdiffusion, non-Gaussian diffusion, and the dynamic heterogeneity in the static disordered medium. It also offers an insight on how the cell would benefit from the quasistatic disordered structures.

Diffusion-limited reactions in dynamic heterogeneous media

Most biochemical reactions in living cells rely on diffusive search for target molecules or regions in a heterogeneous overcrowded cytoplasmic medium. Rapid rearrangements of the medium constantly change the effective diffusivity felt locally by a diffusing particle and thus impact the distribution of the first-passage time to a reaction event. Here, we investigate the effect of these dynamic spatiotemporal heterogeneities onto diffusion-limited reactions. We describe a general mathematical framework to translate many results for ordinary homogeneous Brownian motion to heterogeneous diffusion. In particular, we derive the probability density of the first-passage time to a reaction event and show how the dynamic disorder broadens the distribution and increases the likelihood of both short and long trajectories to reactive targets. While the disorder slows down reaction kinetics on average, its dynamic character is beneficial for a faster search and realization of an individual reaction event triggered by a single molecule.

“Diffusing diffusivity” concept has been recently put forward to account for rapid structural rearrangements in soft matter and biological systems. Here the authors propose a general mathematical framework to compute the distribution of first-passage times in a dynamically heterogeneous medium.

Ergodicity, rejuvenation, enhancement, and slow relaxation of diffusion in biased continuous-time random walks

Bias plays an important role in the enhancement of diffusion in periodic potentials. Using the continuous-time random walk in the presence of a bias, we report on an interesting phenomenon for the enhancement of diffusion by the start of the measurement in a random energy landscape. When the variance of the waiting time diverges, in contrast to the bias-free case, the dynamics with bias becomes superdiffusive. In the superdiffusive regime, we find a distinct initial ensemble dependence of the diffusivity. Moreover, the diffusivity can be increased by the aging time when the initial ensemble is not in equilibrium. We show that the time-averaged variance converges to the corresponding ensemble-averaged variance; i.e., ergodicity is preserved. However, trajectory-to-trajectory fluctuations of the time-averaged variance decay unexpectedly slowly. Our findings provide a rejuvenation phenomenon in the superdiffusive regime, that is, the diffusivity for a nonequilibrium initial ensemble gradually increases to that for an equilibrium ensemble when the start of the measurement is delayed.

Non-self-averaging behaviors and ergodicity in quenched trap models with finite system sizes

Tracking tracer particles in heterogeneous environments plays an important role in unraveling material properties. These heterogeneous structures are often static and depend on the sample realizations. Sample-to-sample fluctuations of such disorder realizations sometimes become considerably large. When we investigate the sample-to-sample fluctuations, fundamental averaging procedures are a thermal average for a single disorder realization and the disorder average for different disorder realizations. Here we report on non-self-averaging phenomena in quenched trap models with finite system sizes, where we consider the periodic and the reflecting boundary conditions. Sample-to-sample fluctuations of diffusivity greatly exceed trajectory-to-trajectory fluctuations of diffusivity in the corresponding annealed model. For a single disorder realization, the time-averaged mean square displacement and position-dependent observables converge to constants because of the existence of the equilibrium distribution. This is a manifestation of ergodicity. As a result, the time-averaged quantities depend neither on the initial condition nor on the thermal histories but depend crucially on the disorder realization.

Non-Gaussian diffusion in static disordered media

Non-Gaussian diffusion is commonly considered as a result of fluctuating diffusivity, which is correlated in time or in space or both. In this work, we investigate the non-Gaussian diffusion in static disordered media via a quenched trap model, where the diffusivity is spatially correlated. Several unique effects due to quenched disorder are reported. We analytically estimate the diffusion coefficient D_{dis} and its fluctuation over samples of finite size. We show a mechanism of population splitting in the non-Gaussian diffusion. It results in a sharp peak in the distribution of displacement P(x,t) around x=0, that has frequently been observed in experiments. We examine the fidelity of the coarse-grained diffusion map, which is reconstructed from particle trajectories. Finally, we propose a procedure to estimate the correlation length in static disordered environments, where the information stored in the sample-to-sample fluctuation has been utilized.

Nonlocality of relaxation rates in disordered landscapes

We investigate both analytically and by numerical simulation the relaxation of an overdamped Brownian particle in a 1D multiwell potential. We show that the mean relaxation time from an injection point inside the well down to its bottom is dominated by statistically rare trajectories that sample the potential profile outside the well. As a consequence, also the hopping time between two degenerate wells can depend on the detailed multiwell structure of the entire potential. The nonlocal nature of the transitions between two states of a disordered landscape is important for the correct interpretation of the relaxation rates in complex chemical-physical systems, measured either through numerical simulations or experimental techniques.

Universal Fluctuations of Single-Particle Diffusivity in a Quenched Environment

Local diffusion coefficients in disordered materials such as living cells are highly heterogeneous. We consider finite systems with quenched disorder in order to investigate the effects of sample disorder fluctuations and confinement on single-particle diffusivity. While the system is ergodic in a single disorder realization, the time-averaged mean square displacement depends crucially on the disorder; i.e., the system is ergodic but non-self-averaging. Moreover, we show that the disorder average of the time-averaged mean square displacement decreases with the system size. We find a universal distribution for diffusivity in the sense that the shape of the distribution does not depend on the dimension. Quantifying the degree of the non-self-averaging effect, we show that fluctuations of single-particle diffusivity far exceed the corresponding annealed theory and also find confinement effects. The relevance for experimental situations is also discussed.

Escape process in systems characterized by stable noises and position-dependent resting times

Stochastic systems characterized by a random driving in a form of the general stable noise are considered. The particle experiences long rests due to the traps the density of which is position dependent and obeys a power-law form attributed to the underlying self-similar structure. Both the one- and two-dimensional cases are analyzed. The random walk description involves a position-dependent waiting time distribution. On the other hand, the stochastic dynamics is formulated in terms of the subordination technique where the random time generator is position dependent. The first passage time problem is addressed by evaluating a first passage time density distribution and an escape rate. The influence of the medium nonhomogeneity on those quantities is demonstrated; moreover, the dependence of the escape rate on the stability index and the memory parameter is evaluated. Results indicate essential differences between the Gaussian case and the case involving Lévy flights.