Random Transition Rates Research Articles

Large deviations for the density and current in non-equilibrium-steady-states on disordered rings

The so-called ‘Level 2.5’ general result for the large deviations of the joint probability of the density and of currents for Markov jump processes is applied to the case of N independent particles on a ring with random transition rates. We first focus on the directed trap model, where the contractions needed to obtain the large deviations properties of the density and of the current alone can be explicitly written in each disordered sample, and where the deformed Markov operator needed to evaluate the generating function of the current can also be explicitly analyzed via its highest eigenvalue and the corresponding left and right eigenvectors. We then turn to the non-directed model, where the tails for large currents of the rate function for the current alone can still be studied explicitly, either via contraction or via the deformed Markov operator method. We mention the differences with the large deviations properties of the Fokker–Planck dynamics on disordered rings.

Phase transition for SIR model with random transition rates on complete graphs

We are concerned with the susceptible-infective-removed (SIR) model with random transition rates on complete graphs C n with n vertices. We assign independent and identically distributed (i.i.d.) copies of a positive random variable ξ on each vertex as the recovery rates and i.i.d. copies of a positive random variable ρ on each edge as the edge infection weights. We assume that a susceptible vertex is infected by an infective one at rate proportional to the edge weight on the edge connecting these two vertices while an infective vertex becomes removed with rate equals the recovery rate on it, then we show that the model performs the following phase transition when at t = 0 one vertex is infective and others are susceptible. There exists λ c > 0 such that when λ < λ c ; the proportion r∞ of vertices which have ever been infective converges to 0 weakly as n → +∞ while when λ > λ c ; there exist c(λ) > 0 and b(λ) > 0 such that for each n ≥ 1 with probability p ≥ b(λ); the proportion r∞ ≥ c(λ): Furthermore, we prove that λ c is the inverse of the production of the mean of ρ and the mean of the inverse of ξ.

Entropy production in systems with random transition rates close to equilibrium.

We study the entropy production of systems out of equilibrium using random networks. We focus on systems with a finite number of states described by a master equation close to equilibrium. The dynamics are mapped into a network of states (nodes) connected by transition rates (links). Using this framework, we analyze the entropy production of ensembles of randomly generated networks owing to specific constraints (e.g., size or symmetries) and identify the most important parameters that determine its value. This analysis gives a null-model estimation for the entropy production that can be used for comparison with specific systems.

Effective-medium approximation for lattice random walks with long-range jumps.

We consider the random walk on a lattice with random transition rates and arbitrarily long-range jumps. We employ Bruggeman's effective-medium approximation (EMA) to find the disorder-averaged (coarse-grained) dynamics. The EMA procedure replaces the disordered system with a cleverly guessed reference system in a self-consistent manner. We give necessary conditions on the reference system and discuss possible physical mechanisms of anomalous diffusion. In the case of a power-law scaling between transition rates and distance, lattice variants of Lévy-flights emerge as the effective medium, and the problem is solved analytically, bearing the effective anomalous diffusivity. Finally, we discuss several example distributions and demonstrate very good agreement with numerical simulations.

Variational approach to the diffusion on inhomogeneous lattices

Abstract General formulation of the variational approach to the collective diffusion process is presented. The method has been primarily invented to deal with problems of many interacting particles at the homogeneous surfaces. It turned out that the approach can be easily generalized to the cases where diffusion occurs at inhomogeneous lattices. It can be applied to describe interacting and noninteracting adatoms. Diffusion of single particle over lattices of various geometry can be also analyzed within this approach. The use of the method in its general formulation is illustrated in the examples of 1D lattices with random transition rates. Simple results of independent particles in the random lattices can be easily reproduced on using variational approach and solutions for more complicated cases including particle–particle correlation can be derived. It is shown that due to the dynamical correlations increase of the diffusion coefficient can be suppressed or generated depending on the concrete system realizations.

Infinite randomness critical behavior of the contact process on networks with long-range connections

The contact process and the slightly different susceptible–infected–susceptible model are studied on long-range connected networks in the presence of random transition rates by means of a strong disorder renormalization group method and Monte Carlo simulations. We focus on the case where the connection probability decays with the distance l as p(l) ≃ βl−2 in one dimension. Here, the graph dimension of the network can be continuously tuned with β. The critical behavior of the models is found to be described by an infinite randomness fixed point which manifests itself in logarithmic dynamical scaling. Estimates of the complete set of the critical exponents, which are found to vary with the graph dimension, are provided by different methods. According to the results, the additional disorder of transition rates does not alter the infinite randomness critical behavior induced by the disordered topology of the underlying random network. This finding opens up the possibility of the application of an alternative tool, the strong disorder renormalization group method to dynamical processes on topologically disordered structures. As the random transverse-field Ising model falls into the same universality class as the random contact process, the results can be directly transferred to that model defined on the same networks.

Steady‐state random walk on connected graph of arbitrary topology with random and non‐symmetric transition rates

Abstractmagnified imageThe general solution and its graphical interpretation for the stationary occupation probability of nodes for a random‐walk on a connected network of arbitrary topology with random and non‐symmetric transition rates are presented. Configurational averaging over the quenched random transition rates is performed for several simple networks: open chains, simple and general stars, two interacting stars. It is demonstrated that disorder in transition rates for a random walker induces variable occupation probability for different nodes in the network depending on their location. The boundary nodes are shown to be occupied with higher probability than the central ones.

Normal and anomalous diffusion in random potential landscapes

A relation between the effective diffusion coefficient in a lattice with random site energies and random transition rates and the macroscopic conductivity in a random resistor network allows for elucidating possible sources of anomalous diffusion in random potential models. We show that subdiffusion is only possible either if the mean Boltzmann factor in the corresponding potential diverges or if the percolation concentration in the system is equal to unity (or both), and that superdiffusion is impossible in our system under any condition. We show also other useful applications of this relation.

Nonexponential relaxations in sensor arrays: forecasting strategy for electronic nose performance

The ability to select definite and unique selectivity profile from a variety of molecular organic materials predetermines their using as sensitive layers for cross-selective sensor massifs. However, to gain greater insight into how the instrumental performance depends on various parameters of such sensitive layers there is a need for a flexible, yet cost effective methodology for the estimation of the natural recourses of organic materials to monitor complex gas mixtures. In the present work the use of the cluster analysis method in the ‘fuzzy logic’ concept for the optimization of the cross-selective sensor arrays (‘electronic nose’, EN) is considered. This approach enables to purposefully form the sensor elements arrays with definite chemical functionality optimized for the solution of the specific applied problems. The criteria of the optimization of the sensor response, number and type of the sensor elements are considered with the goal of the improvement of the classification of the pharmaceutical products of the wide use. On the base of analysis of the response kinetic peculiarities the physical mechanisms were considered, which determines the peculiarities of the adsorption–desorption processes at the interface. It was shown that adsorption of molecules from complex mixtures decays nonexponentially in general: in a “weak” adsorbers films it is exponential and approaches a stretched exponential for “strong” adsorbers film. Moreover the stretched exponentiality in adsorption–desorption processes can be described in terms of probabilistic model of relaxation based on the self-similarity of distributed random transition rates. We have demonstrated the first application of the stretched exponential low for nonexponential relaxations of QCM sensors with organic sensitive layers. There is a good agreement between molecular structure of materials, interfacial dynamics of sensors and discrimination ability of massifs thereon.

Diffusion-limited reaction in the presence of random fields and transition rates

The diffusion-limited reaction was studied on a one-dimensional lattice in the presence of random fields and transition rates using Monte Carlo simulations. In the case of transition rates the hopping probabilities at a site are distributed according to the power law p(y)=νyν−1 with 0&amp;lt;ν⩽1 and 0&amp;lt;y⩽1. The density of the reactants decays according to a power-law, C(t)∼t−α(ν) for A+A→0 and A+B→0 annihilation reactions. The exponent α(ν) depends on the disorder exponent ν. For A+A→0, we found α(ν)=ν/(1+ν). For A+B→0, we observed α=0.25 at ν&amp;gt;0.4 and α decreases monotonically for ν&amp;lt;0.4. In the case of the random fields the density decays according to C(t)∼[b(E)/log(t)]2 regardless of the strength of the random fields E for A+A→0 and A+A→A reactions, where b(E)∼log[(1+E)/(1−E)]. The diffusion-limited coagulation A+A→A belongs to the same universality class as the A+A→0 reaction under the random fields. For A+B→0 annihilations we observe that the density decays according to C(t)∼b(E)/log(t) in the presence of the random fields.

Tiger and Rabbits: a single trap and many random walkers

We study a one-dimensional system with a single trap (Tiger) initially located at the origin, and many random-walkers (Rabbits) initially uniformly distributed throughout the infinite or the semi-infinite space. For a mobile imperfect trap, we study the spatiotemporal properties of the system, such as the trapping rate, the particle distribution and the segregation around the trap, all as a function of the diffusivities of both the trap and the walkers. For a static trap, we present results of various measures of segregation, in particular on a few types of disordered chains, such as random local bias fields (the Sinai model) and random transition rates.

Taylor dispersion of particles in stratified media with random transition rates

The dispersion of particles in a stratified medium with a flow normal to the stratification is studied for the case when the transition rates between the various strata are random variables. One- and two-dimensional stratified systems are considered. The effective diffusion coefficient in the flow direction is computed numerically. The numerical results are compared with the predictions of a mean-field theory known as the effective (or coherent) medium approximation.

Diffusion Coefficients of Single and Many Particles in Lattices with Different Forms of Disorder

A survey is given on asymptotic diffusion coefficients of particles in lattices with random transition rates. Exact and approximate results for single particles are reviewed. A recent exact expression in $d = 1$ which includes occupation factors is discussed. The utilization of the result is demonstrated for the Miller-Abrahams model and a model of random barriers combined with random traps. Exact and approximate results for the site-exclusion model in disordered lattices are also given.

Random transition rate model of stretched exponential relaxation

Stretched exponential regression to equilibrium is a frequently observed phenomenon in relaxation of non-equilibrium states. The question of the origins of the stretched exponentiality is addressed in terms of the probabilistic model of relaxation, based on the self-similarly distributed random transition rates. Each rate corresponds to one channel of relaxation and channels are assumed to operate in a parallel way, i.e., individual relaxation events are independent. As a consequence the effective transition rate obtained as a normalized sum of individual rates is found to be distributed according to the (asymmetric) Levy stable distribution, which is known to be a necessary and sufficient condition for stretched exponential relaxation. This known result is restated now within the framework of a model, which has the simple phenomenology of the parallel channels, but which however operates with self-similar dynamics. Moreover, the derivations are entirely carried out in terms of characteristic functions of untransformed random variables. The model closely resembles the existing probabilistic models and the differencies are mainly found in the way to motivate the self-similarity of dynamics and in different emphasis on the starting assumptions. The main motivation has been to point out the inherent relatedness of all probabilistic models operating with only one self-similar stochastic process, and to suggest that relaxation can be handled with a single class of well defined functions.

Anomalous segregation at a single trap in disordered chains.

We study the repulsion of Brownian particles induced by a single trap on disordered chains. Two different types of disorder are considered: random local bias fields (the Sinai model) and random transition rates. We discuss two possible measures of segregation for each system, and show that they can have either similar or different universal behavior, depending on the properties of diffusion subject to the hard-core potential in each system. We also report on anomalous scaling properties for the average trapping rate and the average density profile of the diffusing particles. It is surprisingly shown that the latter is not spatially linear in the vicinity of the trap, but rather has a flat tail in the case of random fields, and a nonuniversal power-law in the case of random transition rates.

Fluctuations of the probability density of diffusing particles for different realizations of a random medium.

We study the fluctuations of the probability density P(r,t) of diffusing particles to be at distance r at time t in the presence of random potentials, represented by random transition rates. We find an exact relation which expresses all the moments of P(r,t) in terms of its first moment, for both quenched and annealed disorder and for any dimension. From this relation follows that anomalous diffusion implies nontrivial behavior of the moments of P(r,t), such as an exponential divergence of the relative fluctuations for large r.

Logarithmic correction to fractal time behavior for diffusion in ultrametric spaces with random energy barriers

A new model for the diffusion in ultrametric spaces with random transition rates is suggested. The ultrametric structure is generated by a branching process with a random number of branches, whereas the randomness of the transition rates is described in terms of an exponential distribution of activation energies. The interaction between these two random processes is analyzed. We show that the contribution of the ultrametric structure outweighs the contribution of the random energy barriers. The randomness of the energy barriers generates a logarithmic correction for large time. The distribution function ψ(t) of the waiting time in a given state has the following asymptotic behavior: ψ(t) ∼ t-1 · (ln vt)- (H + 1) Λ(ln ln vt) as t → ∞ where 1 ⩾ H > 0 is an exponent related to the ultrametric structure, v is the maximum jump frequency and Λ(ln ln vt) is a periodic function of ln ln vt.

Segregation at a Single Trap in the Presence of Fields

AbstractThere have been a number of recent investigations of segregation properties of diffusing particles in the presence of a single static trap in low dimensions. We study these properties when the diffusing particles are subject to different forms of external fields: global constant bias, random bias fields (Sinai model) and random transition rates. We discuss two measures of segregation, the distances from the trap either to the point at which the concentration profile reaches a specified fraction of its bulk value, or to the nearest unreacted particle. For the cases of global bias (both away from, and towards the trap) and random fields, we found that both measures of segregation have the same asymptotic temporal behavior, while for random transition rates they differ. We explain this difference by relating the nearest-neighbor distance measure to properties of hard-core diffusion in these systems. We also found anomalous spatial shapes for the profile in the vicinity of the trap in the random systems, as well as anomalous reaction rates.

Directed random walks in random environments

We use holding time methods to study the asymptotic behavior of pure birth processes with random transition rates. Both the normal and slow approaches to infinity are studied. Fluctuations are shown to obey the central limit theorem for almost all sample-transition rates. Our results are stronger, and our proofs simpler, then those of recently published studies.

Probability densities of random walks in random systems

We review recent results for the probability distribution of random walkers in random systems, where diffusion is anomalous and the mean-square displacement scales with time as R2(t)∼t2 w, dw > 2. The random systems are characterized by structural disorder and by random transition rates. In general, the mean distribution function 〈P(r, t)〉 of the random walkers is a stretched Gaussian and scales as log[P(r,t)P(r,0)]∼−[rR(t)]u, where u=dw(dw−1). On random fractals, the fluctuations of the density distribution P(r, t), for fixed distance r and time t, have a broad logarithmic distribution. The average moments 〈Pq〉 scale in a multifractal way as 〈P〉τ(q), where τ(q)∼qτ,γ<1. In contrast, in chemical l-space the fluctuations of P are narrow and 〈Pq〉∼〈P〉q.