Continuous Time Random Walk Description Research Articles

Diffusion through Nanopores in Connected Lipid Bilayer Networks.

A biomimetic model of cell-cell communication was developed to probe the passive molecular transport across ion channels inserted in synthetic lipid bilayers formed between contacting droplets arranged in a linear array. Diffusion of a fluorescent probe across the array was measured for different pore concentrations. The diffusion characteristic timescale is found to vary nonlinearly with the pore concentration. Our measurements are successfully modeled by a continuous time random walk description whose waiting time is the first exit time from a droplet through a cluster of pores. The size of the cluster of pores is found to increase with their concentration. Our results provide a direct link between the mesoscopic permeation properties and the microscopic characteristics of the pores, such as their number, size, and spatial arrangement.

Diffusion in Nanopore Connected Bilayer Networks

A biomimetic model of cell-cell communication was developed to probe the passive molecular transport across ion channels inserted in synthetic lipid bilayers formed between contacting droplets arranged in a linear array. Diffusion of a fluorescent probe across the array was measured for different pore concentrations. The diffusion characteristic time scale is found to vary non-linearly with the pore concentration. Our measurements are successfully modeled by a continuous time random walk description, whose waiting time is the first exit time from a droplet through a cluster of pores. The size of the cluster of pores is found to increase with their concentration. Our results provide a direct link between the mesoscopic permeation properties and the microscopic characteristics of the pores such as their number, size and spatial arrangement.

Cage-jump motion reveals universal dynamics and non-universal structural features in glass forming liquids

.The sluggish and heterogeneous dynamics of glass forming liquids is frequently associated to the transient coexistence of two phases of particles, respectively with a high and low mobility. In the absence of a dynamical order parameter that acquires a transient bimodal shape, these phases are commonly identified empirically, which makes it difficult to investigate their relation with the structural properties of the system. Here we show that the distribution of single particle diffusivities can be accessed within a continuous time random walk description of the intermittent motion, and that this distribution acquires a transient bimodal shape in the deeply supercooled regime, thus allowing for a clear identification of the two coexisting phases. In a simple two-dimensional glass forming model, the dynamic phase coexistence is accompanied by a striking structural counterpart: the distribution of the crystalline-like order parameter becomes also bimodal on cooling, with increasing overlap between ordered and immobile particles. This simple structural signature is absent in other models, such as the three-dimesional Kob–Andersen Lennard-Jones mixture, where more sophisticated order parameters might be relevant. In this perspective, the identification of the two dynamical coexisting phases opens the way to deeper investigations of structure-dynamics correlations.

Diffusion and Subdiffusion of Interacting Particles on Comblike Structures.

We study the dynamics of a tracer particle (TP) on a comb lattice populated by randomly moving hard-core particles in the dense limit. We first consider the case where the TP is constrained to move on the backbone of the comb only. In the limit of high density of the particles, we present exact analytical results for the cumulants of the TP position, showing a subdiffusive behavior ∼t^{3/4}. At longer times, a second regime is observed where standard diffusion is recovered, with a surprising nonanalytical dependence of the diffusion coefficient on the particle density. When the TP is allowed to visit the teeth of the comb, based on a mean-field-like continuous time random walk description, we unveil a rich and complex scenario with several successive subdiffusive regimes, resulting from the coupling between the geometrical constraints of the comb lattice and particle interactions. In this case, remarkably, the presence of hard-core interactions asymptotically speeds up the TP motion along the backbone of the structure.

Life and Death of Stationary Linear Response in Anomalous Continuous Time Random Walk Dynamics

Linear theory of stationary response in systems at thermal equilibrium requires to find equilibrium correlation function of unperturbed responding system. Studies of the response of the systems exhibiting anomalously slow dynamics are often based on the continuous time random walk description (CTRW) with divergent mean waiting times. The bulk of the literature on anomalous response contains linear response functions like one by Cole-Cole calculated from such a CTRW theory and applied to systems at thermal equilibrium. Here we show within a fairly simple and general model that for the systems with divergent mean waiting times the stationary response at thermal equilibrium is absent, in accordance with some recent studies. The absence of such stationary response (or dying to zero non-stationary response in aging experiments) would confirm CTRW with divergent mean waiting times as underlying physical relaxation mechanism, but reject it otherwise. We show that the absence of stationary response is closely related to the breaking of ergodicity of the corresponding dynamical variable. As an important new result, we derive a generalized Cole-Cole response within ergodic CTRW dynamics with finite waiting time. Moreover, we provide a physically reasonable explanation of the origin and wide presence of 1/f noise in condensed matter for ergodic dynamics close to normal, rather than strongly deviating.

Quench, Equilibration, and Subaging in Structural Glasses

In the glassy state, all materials undergo a process of structural recovery as they age towards equilibrium. The resultant increase of relaxation times t(α) is frequently described with a sublinear power of the wait time t(w)(μ) with an apparent aging exponent μ. We show with molecular dynamics simulations of a Lennard-Jones glass former at various temperatures that the observed aging exponent can be strongly influenced by crossover effects from the freshly quenched state at short t(w) and into the equilibrated state at long t(w). The aging behavior on the molecular level is quantitatively reproduced by a coarse-grained continuous time random walk description over the entire range of temperatures and wait times. Our model glass always shows normal aging, t(α)∼t(w), when the observation time window is no longer affected by crossover effects, in agreement with the well-known trap model of aging.

Survival probability of a particle in a sea of mobile traps: A tale of tails

We study the long-time tails of the survival probability P(t) of an A particle diffusing in d-dimensional media in the presence of a concentration rho of traps B that move subdiffusively, such that the mean square displacement of each trap grows as tgamma with 0 < or = gamma < or =1. Starting from a continuous time random walk description of the motion of the particle and of the traps, we derive lower and upper bounds for P(t) and show that for gamma < or =2/(d+2) these bounds coincide asymptotically, thus determining asymptotically exact results. The asymptotic decay law in this regime is exactly that obtained for immobile traps. This means that for sufficiently subdiffusive traps, the moving A particle sees the traps as essentially immobile, and Lifshitz or trapping tails remain unchanged. For gamma >2/(d+2) and d< or =2 the upper and lower bounds again coincide, leading to a decay law equal to that of a stationary particle. Thus, in this regime the moving traps see the particle as essentially immobile. For d>2 , however, the upper and lower bounds in this gamma regime no longer coincide, and the decay law for the survival probability of the A particle remains ambiguous.

The continuous‐time random walk description of the non‐equilibrium mechanical response of crosslinked elastomers

AbstractThe application of the continuous‐time random walk model to the description of the non‐equilibrium stress response of a crosslinked network is discussed. Different versions of the continuous‐time random walk model lead to the prediction of a stress relaxation function having either a fractional‐power law form or a fractional‐exponential law form. Both forms are tested against experimental data on the stress relaxation and loss modulus behaviours of crosslinked natural rubber.

On the frequency dependence of the conductivity in random walk models with internal states

The random walk of a particle in two models of an ideal lattice with internal states has been studied. In the first model one vertical ladder of energy levels exists at each lattice site, in the second model different ladders of levels lead to neighboring sites. In thermal equilibrium, the mobility of the particle was found frequency-independent in the first model and frequency-dependent in the second. The continuous-time random walk (CTRW) description of both models has been investigated. The waiting-time distribution for the first case has been obtained analytically in the Laplace domain. An example for φ (t) has been calculated numerically by inverse Laplace transformation. In the second case an extension of the CTRW description to a backward jump model has been performed and the corresponding waiting-time distributions have been calculated.

On the equivalence between multistate-trapping and continuous-time random walk models

We compare two model descriptions of transport in a material with traps. We establish the equivalence between the n-state trapping model where equilibrium initial conditions are used, and the continuous-time random walk description where a distinct first-waiting-time distribution is introduced. Tunaley's result on the frequency independence of the ac conductivity is corroborated by this equivalence.