Stochastic Transport in a Disordered Solid. I. Theory

Scher, H.; Lax, M.

doi:10.1103/physrevb.7.4491

Cited by 1,174 publications

(678 citation statements)

References 26 publications

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“…(34), (45) and (54)] for a stationary initial condition and D ν [Eqs. (32), (43) and (53)] if the system is not initially in the stationary state (see also Figs. 2, 3 and 5).…”

Section: Discussionmentioning

confidence: 99%

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Scaling Green-Kubo Relation and Application to Three Aging Systems

et al. 2014

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The Green-Kubo formula relates the spatial diffusion coefficient to the stationary velocity autocorrelation function. We derive a generalization of the Green-Kubo formula that is valid for systems with longrange or nonstationary correlations for which the standard approach is no longer valid. For the systems under consideration, the velocity autocorrelation function hvðt þ τÞvðtÞi asymptotically exhibits a certain scaling behavior and the diffusion is anomalous, hx 2 ðtÞi ≃ 2D ν t ν . We show how both the anomalous diffusion coefficient D ν and the exponent ν can be extracted from this scaling form. Our scaling GreenKubo relation thus extends an important relation between transport properties and correlation functions to generic systems with scale-invariant dynamics. This includes stationary systems with slowly decaying power-law correlations, as well as aging systems, systems whose properties depend on the age of the system. Even for systems that are stationary in the long-time limit, we find that the long-time diffusive behavior can strongly depend on the initial preparation of the system. In these cases, the diffusivity D ν is not unique, and we determine its values, respectively, for a stationary or nonstationary initial state. We discuss three applications of the scaling Green-Kubo relation: free diffusion with nonlinear friction corresponding to cold atoms diffusing in optical lattices, the fractional Langevin equation with external noise recently suggested to model active transport in cells, and the Lévy walk with numerous applications, in particular, blinking quantum dots. These examples underline the wide applicability of our approach, which is able to treat very different mechanisms of anomalous diffusion.

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“…(34), (45) and (54)] for a stationary initial condition and D ν [Eqs. (32), (43) and (53)] if the system is not initially in the stationary state (see also Figs. 2, 3 and 5).…”

Section: Discussionmentioning

confidence: 99%

“…Our final example is a paradigm model for aging, the Lévy walk [30,31], which is an extension of the continuous-time random walk [32,33] to superdiffusive dynamics. In its simplest form, this model describes a particle whose velocity can have values AEv 0 with random switching between them.…”

Section: Introductionmentioning

confidence: 99%

Scaling Green-Kubo Relation and Application to Three Aging Systems

et al. 2014

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“…where f(m) describes the initial distribution of the charge, a is the distance between neighboring monomer units, (n À m)a is the distance between the orbitals localized on monomer units n and m, and c n (t,m) is the coefficient of the orbital on monomer unit n at time t for the case of the hole wavefunction being initially localized on monomer unit m. According to the work of Kubo, the frequency-dependent (one-dimensional) mobility of charge carriers is given by [126][127][128] …”

Section: This Journal Is C the Owner Societies 2010mentioning

confidence: 99%

“…An implicit convergence factor exp(Àot) (lim o -0) is understood in the integral. 127,128 For normal Gaussian diffusion the mean squared displacement of charge carriers moving along an infinitely long one-dimensional chain increases linearly with time: View Online where D is the diffusion constant. In this special case, the mobility is frequency independent, and eqn (27) reduces to the Einstein relation…”

Section: This Journal Is C the Owner Societies 2010mentioning

confidence: 99%

Single molecule charge transport: from a quantum mechanical to a classical description

Kocherzhenko

Grozema

Siebbeles

2011

Phys. Chem. Chem. Phys.

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This paper explores charge transport at the single molecule level. The conductive properties of both small organic molecules and conjugated polymers (molecular wires) are considered. In particular, the reasons for the transition from fully coherent to incoherent charge transport and the approaches that can be taken to describe this transition are addressed in some detail. The effects of molecular orbital symmetry, quantum interference, static disorder and molecular vibrations on charge transport are discussed. All of these effects must be taken into account (and may be used in a functional way) in the design of molecular electronic devices. An overview of the theoretical models employed when studying charge transport in small organic molecules and molecular wires is presented.

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“…Anomalous behavior characterized through constant time periods (called also trapping events) is observed in variety of physical systems, including charge carrier transport in amorphous semiconductors [29,28,25], transport in micelles [24], intracellular transport [4], motion of mRNA molecules inside E. coli cells [9], for a review including discussion of different applications see [5]. This specific behavior is also typical for some financial data especially corresponding to interest rates and stock prices, for which the constant time periods occur when the liquidity of the analyzed assets is low [12].…”

Section: Introductionmentioning

confidence: 99%

Calibration of the Subdiffusive Arithmetic Brownian Motion with Tempered Stable Waiting-Times

Orzeł

Wyłomańska

2011

J Stat Phys

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In the classical analysis many models used to real data description are based on the standard Brownian diffusion-type processes. However, some real data exhibit characteristic periods of constant values. In such cases the popular systems seem not to be applicable. Therefore we propose an alternative approach, based on the combination of the popular arithmetic Brownian motion and tempered stable subordinator. The probability density function of the proposed model can be described by a Fokker-Planck type equation and therefore it has many similar properties as the popular arithmetic Brownian motion. In this paper we propose the estimation procedure for the considered tempered stable subdiffusive arithmetic Brownian motion and calibrate the analyzed process to the real financial data.

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Stochastic Transport in a Disordered Solid. I. Theory

Cited by 1,174 publications

References 26 publications

Scaling Green-Kubo Relation and Application to Three Aging Systems

Scaling Green-Kubo Relation and Application to Three Aging Systems

Single molecule charge transport: from a quantum mechanical to a classical description

Calibration of the Subdiffusive Arithmetic Brownian Motion with Tempered Stable Waiting-Times

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