Continuous-time Random-walk Approximation Research Articles

Skorokhod Reflection Problem for Delayed Brownian Motion with Applications to Fractional Queues

Several queueing systems in heavy traffic regimes are shown to admit a diffusive approximation in terms of the Reflected Brownian Motion. The latter is defined by solving the Skorokhod reflection problem on the trajectories of a standard Brownian motion. In recent years, fractional queueing systems have been introduced to model a class of queueing systems with heavy-tailed interarrival and service times. In this paper, we consider a subdiffusive approximation for such processes in the heavy traffic regime. To do this, we introduce the Delayed Reflected Brownian Motion by either solving the Skorohod reflection problem on the trajectories of the delayed Brownian motion or by composing the Reflected Brownian Motion with an inverse stable subordinator. The heavy traffic limit is achieved via the continuous mapping theorem. As a further interesting consequence, we obtain a simulation algorithm for the Delayed Reflected Brownian Motion via a continuous-time random walk approximation.

Negative differential mobility of weakly driven particles in models of glass formers

We study the response of probe particles to weak constant driving in kinetically constrained models of glassy systems, and show that the probe's response can be nonmonotonic and give rise to negative differential mobility: increasing the applied force can reduce the probe's drift velocity in the force direction. Other significant nonlinear effects are also demonstrated, such as the enhancement with increasing force of the probe's fluctuations away from the average path, a phenomenon known in other contexts as giant diffusivity. We show that these results can be explained analytically by a continuous-time random walk approximation where there is decoupling between persistence and exchange times for local displacements of the probe. This decoupling is due to dynamic heterogeneity in the glassy system, which also leads to bimodal distributions of probe particle displacements. We discuss the relevance of our results to experiments.

Polarization dynamics and formation of polar nanoregions in relaxor ferroelectrics

Polarization dynamics and formation of polar nanoregions (PNR) in relaxor ferroelectrics is considered within a model of interacting short range polar clusters formed by off-center ions in highly polarizable materials. The model is applicable on the mesoscopic level and takes explicitly into account the distribution of cluster relaxation times and the existence of quenched random fields which control the size of PNR. Using self-consistent random field theory and continuous time random walk approximation, a relationship is established between the average polarization dynamics contributing to the low frequency dielectric response, and the local polarization dynamics determining NMR spin lattice relaxation time [Blinc et al., Phys. Rev. B 63, 024104 (2001); Cordero et al., ibid. 71, 094112 (2005)]. The lengthscale of PNR estimated from the obtained universal relationship between the parameters of the soft mode dispersion curve and the static dielectric permittivity is in agreement with the experiment [Vakhrushev et al., Physica B 156-157, 90 (1989)]. The predicted proportionality, between the intensity of diffuse neutron scattering from dynamic PNR and square of PNR correlation length, which has been recently a subject of controversy, is also in agreement with experiment by Vakhrushev et al. Therefore, we conclude that the model captures essential features of the static and dynamic behavior of relaxor ferroelectrics, and could be used for the characterization of new relaxor materials.

Chemistry on interstellar grains

In this review, our current knowledge of the chemistry that occurs on cold interstellar grain surfaces is examined. The formation of molecular hydrogen from hydrogen atoms is emphasized. Various methods of modeling diffusive reactions on grain surfaces are illustrated, including the rate equation method, and two basic stochastic approaches. The results of a detailed Monte Carlo method - the continuous-time random-walk approximation - in which the problem of random walk in two dimensions is treated exactly, are shown and analyzed for the case of H2 formation. Both homogeneous surfaces such as olivine and amorphous carbon and inhomogeneous surfaces are studied. Results from this method show that (a) inhomogeneous surfaces allow efficient H2 formation over wider temperature ranges than do homogeneous surfaces, and (b) the reaction efficiency declines for grains smaller than a certain size, but the effect is smaller for inhomogeneous surfaces.

Time dependent conductivity in disordered systems

The time dependent current in disordered systems under a step applied voltage for a planar synmetry is deduced according to the continuous time random walk approximation. Known dielectric response functions like Cole-Cole, Davidson-Cole, Havriliak-Negami and a few others are used as hopping time distribution functions in order to generate conductive responses. A theoretical relation exists between the dielectric and the conductive response which is the same one prevailing between the time derivative of the creep and the relaxation function, as found long ago by Gross (J.Appl.Phys.,18, 212, (1947)). A truncated version of the Widder method, in connection with MapleTM software facilities, was employed to obtain graphical primitives of Laplace transforms.

Space-charge transport in disordered media

The general equations ruling the transport of free excess space charge in disordered media within the continuous-time random walk approximation are inferred from previous works. The equations are shown to have the same structure as those in nondisordered media (for which a mobility may be defined) when written in the Laplace image space. Therefore, the solution in disordered media may be derived from the corresponding solution in nondisordered media if the same initial and boundary conditions are kept. Assuming the extreme dispersive hyperbolic decay of Scher and Montroll, expressions are given for the initial and final stages of the following processes: (1) surface potential decay of corona charged samples, (2) potential buildup for samples charged with a constant current, and (3) the current resulting from charge injection from an ohmic contact. For case (1) it is found that the time dependence of the time derivative of the potential is the same, in absolute value, as in a dispersive photocurrent signal, as far as those extreme stages are concerned.

Random walk approach for the proper estimation of density of defect states in amorphous chalcogenides

Recently, the present authors proposed a correlated barrier hopping model for bipolarons based on continuous-time random-walk approximation to explain ac conductivity in a-As 2Se 3 and estimated the density of defect states based on this approach. It was observed that the density of defect states deduced from this approach is less than as deduced from correlated barrier hopping based on pair approximation and are consistent with the estimated density of defect states from other studies, namely, light-induced electron spin resonance and drift mobility. Now, this approach is extended to the single- and bi-polaron contribution to ac conductivity and apply to the experimental data of a-Se and a-As 2Se 3 + 0.5 at.% Ag. The present approach explains the single- as well as bi-polaron contribution to ac conductivity and the deduced density of defect states is less than that deduced from correlated barrier hopping based on pair approximation.

Estimation of density of charged defects in amorphous chalcogenides from a.c. conductivity: Random-walk approach for bipolarons based on correlated barrier hopping

Abstract Although the pair approximation (PA) based on correlated bamer hopping (CBH) is now fairly well established to explain the a.c. conductivity of amorphous chalcogenides, the density of charged states deduced from PA is large compared with those estimated from other studies, namely light-induced electron spin resonance and drift mobility. We have used the continuous-time random-walk approximation based on CBH to estimate the density of charged defects and have applied it to experimental data for amorphous As2Se3, and a density of charged defects consistent with other measurements has been estimated.

A.c. conduction mechanism of polyacenic semiconductive materials

The real part of a.c.conductivity of the pristine and the BF 4 --doped polyacenic semiconductive (PAS) materials prepared by pyrolytic treatment of phenol-formaldehyde resin has been investigated to analyze the a.c. conduction mechanism in these materials. Dependence of this quantity on the frequency (10 1–10 7 Hz) and the temperature (100–330 K) was examined in detail. A series of measurements has revealed that in the pristine PAS samples the carriers are hopping through the quantum-mechanical tunneling process between the spatial sites represented by the localized states near the Fermi level and that the continuous-time random-walk approximation is applicable to the BF 4 --doped PAS samples.

On the mechanism of d.c. and a.c. transport in transition metal oxide glasses

Abstract Although it is well established that small polarons are formed in transition metal oxide (TMO) glasses, no satisfactory quantitative analysis of d.c. and a.c. transport exists. The temperature dependences of the d.c. and a.c. conductivities for TMO glasses are re-examined. The charge transport cannot be explained quantitatively by the small-polaron model (strong electron-lattice coupling). An alternative possibility—a multiphonon tunnelling process with weak coupling—is also discussed. The a.c. conductivity, which is strongly correlated to the d.c. conductivity and which has never been explained by current theories based on the pair approximation, is interpreted in terms of the continuous-time random-walk approximation.

Analysis of ac conduction in disordered solids

Dyre [J. Appl. Phys. 64, 2456 (1988)] has recently stated that the random free-energy barrier model of ac conduction in disordered solids, solved in the continuous time random-walk approximation with the effects of the maximum jump frequency eliminated, is quantitatively satisfactory in describing hopping conduction for a large number of solids. Here, predictions of this model, equivalent to the long-used box model, which posits a distribution of equally probable activation energies, are examined in depth, both without and with an upper cutoff. It is first demonstrated that the type of log-log plot on which Dyre appears to base his conclusion of quantitative adequacy does not allow adequate discrimination to be made between box-model predictions and those of other models, such as the Kohlrausch–Williams–Watts model, even when exact data are used. The results of numerous complex nonlinear least-squares fits of exact box-model data, and of such data containing substantial proportionally added random errors, to the box model, the WW model, the constant-phase-element model, and the Davidson–Cole [J. Chem. Phys. 19, 1484 (1951)] response model make it clear that when using this fitting technique, one can identify the correct model, discriminate against incorrect ones, and obtain good parameter value estimates for the correct model. Further, when the highest frequency of the data exceeds the maximum jump frequency, its value can be accurately estimated. It is concluded that the case for the quantitative adequacy of the box model remains unproven. Future data fitting using complex nonlinear least squares should, however, allow a best-fit model to be selected unambiguously from those compared.

Multiphonon hopping of electrons on defect clusters in amorphous germanium.

A high density of microvoids could cause inhomogeneous distribution of deep localized states in amorphous germanium. The electronic transport in such media is no longer dominated by Mott variable-range hopping. The dc conductivity is proportional to ${T}^{n}$ with n\ensuremath{\simeq}6, suggesting that the transport is by a multiphonon process with weak electron-lattice coupling. The ac conductivity is explained by the Scher-Lax continuous-time random-walk approximation, producing information on the hopping rate.

Hopping transport of localized pi electrons in amorphous carbon films.

The temperature dependence of the dc and ac conductivities has been studied in sputtered arnorphous carbon films. The dc conductivity is proportional to T with n =15— 17 below room temperature. The mechanism is discussed in terms of the mobility-edge conduction, variable-range hopping, and small-polaron hopping. Quantitatively, these models could not interpret the observed behaviors. Instead, the multiphonon tunneling of localized electrons with weak electron-lattice coupling is suggested to be the dominant transport mechanism. This may be attributed to the ~-bonded nature of sputtered amorphous carbon. The ac conductivity is strongly correlated to the dc conductivity, which has never been explained by the current theories based on the pair approximation. Alternatively, the continuous-time random-walk approximation is shown to be a useful approach. I. INTRODUCTION It is believed that amorphous carbon (a-C) prepared by evaporation or sputtering is predominantly sp bonded with an optical gap of 0.4— 0.7 eV. ' At a carbon sp site, there are three strong o. bonds and one weak m bond lying normal to the o. bonding plane. The ~ states will form both the valence- and conduction-band states. ' The sp sites must be clustered together in aromatic units. The large size of such clusters could produce localized states near the Fermi level E+. The eFect of disorder in a ~-electron system is therefore of particular interest. The present authors have shown that the temperature dependence of the dc conductivity in sputtered a-C is empirically described by o. d, =o.oT, with n =15— 17, which may suggest that the nonpolaronic multiphonon tunneling of localized m electrons (weak coupling with lattice) dominates charge transport. Measurement of ac conductivity could provide direct information about the hopping rate of localized electrons. In the present study, the temperature dependences of both dc and ac conductivities are reported. Possible transport mechanisms, mobility-edge conduction, variable-range hopping (single-phonon process), and multiphonon hopping with strongly (small-polaron) and weakly coupled states, are discussed. It is suggested that the conventional mobility-edge conduction or small-polaron conduction could not explain the overall features of transport in a-C. It is also found that the ac conductivity is strongly correlated with the dc conductivity. This feature, which has never been explained by the current theory based on the pair approximation, is interpreted well in terms of the continuous-time random-walk (CTRW) approximation.

Random walks on random lattices with traps

A critical examination is presented of the continuous time random walk (CTRW) approximation and of frequency-dependent effective transition rate methods for calculating the configurational average of the Laplace transform of the probabilityPs, t|0) that a particle performing a random walk will be at site s timet after it reached the site s0. Some exact results are derived for the form of P(s, t|s0) at long times, and these indicate that the effective transition rate methods are the better approximation for systems with symmetric effective hopping rates, while the CTRW approximation is better for systems containing traps, i.e., states that are much easier to enter than to leave. The implications of these results for calculations of transient currents and of the ac conductivity for amorphous semiconductors are discussed.

A coherent site approximation for transport in disordered systems

For disordered systems in which the transition rate from a site to its neighbours depends only on the properties of the initial site, a coherent site approximation (CSA) is proposed as the appropriate homomorphic cluster coherent potential approximation. This CSA is ideally suited for treating systems in which transitions between localised states of different energy take place via excitation to a band of extended states. In the limits of very short and very long times, the results of the CSA coincide with those of the familiar continuous time random walk approximation and so provide a formal justification for the latter.

An analysis and extension of the continuous time random walk method. I. Theory

The usual continuous time random walk (CTRW) method for analysing the motion of particles between discrete localised states in disordered systems is fairly simple to apply but involves an averaging procedure whose accuracy is unknown. The Laplace transform of the rate equations for such a system is expressed in matrix form, and it is shown that the usual CTRW approximation corresponds to a certain decoupling of their configurational average. This is the first of a hierarchy of CTRW(n) approximations, which treat exactly transitions involving n steps, and conditions are established for the convergence of this series of approximations.