Directed transport in periodically rocked random sawtooth potentials

Denisov, S. I.; Lyutyy, T. V.; Denisova, E. S.; Hänggi, Peter; Kantz, Holger

doi:10.1103/physreve.79.051102

Cited by 9 publications

(9 citation statements)

References 26 publications

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“…In those systems it is the properly tailored periodic "potential landscape" which enforces a conversion of a homogenous stochastic process (Brownian motion for reference) into the directed motion of particles at nanometer scales. That is closely related to the problem of so-called sorting in periodic potentials [29]. Other problem to be addressed concerns ultracold atoms in optical lattices subject to random potentials [30], which might promising not only from a purely scientific point of view, but also with prospects for many technological applications.…”

Section: Discussionmentioning

confidence: 99%

Lévy flights in confining environments: Random paths and their statistics

Żaba

Garbaczewski

Stephanovich

2013

Physica A: Statistical Mechanics and its Applications

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We analyze a specific class of random systems that are driven by a symmetric Lévy stable noise. In view of the Lévy noise sensitivity to the confining "potential landscape" where jumps take place (in other words, to environmental inhomogeneities), the pertinent random motion asymptotically sets down at the Boltzmann-type equilibrium, represented by a probability density function (pdf) ρ * (x) ∼ exp[−Φ(x)]. Since there is no Langevin representation of the dynamics in question, our main goal here is to establish the appropriate path-wise description of the underlying jump-type process and next infer the ρ(x, t) dynamics directly from the random paths statistics. A priori given data are jump transition rates entering the master equation for ρ(x, t) and its target pdf ρ * (x). We use numerical methods and construct a suitable modification of the Gillespie algorithm, originally invented in the chemical kinetics context. The generated sample trajectories show up a qualitative typicality, e.g. they display structural features of jumping paths (predominance of small vs large jumps) specific to particular stability indices µ ∈ (0, 2).

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Section: Discussionmentioning

confidence: 99%

Lévy flights in confining environments: Random paths and their statistics

Żaba

Garbaczewski

Stephanovich

2013

Physica A: Statistical Mechanics and its Applications

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“…In those systems, it is the properly tailored periodic "potential landscape" which enforces a conversion of a homogenous stochastic process (Brownian motion for reference) into the directed motion of particles at nanometer scales. It is closely related to the problem of so-called sorting in periodic potentials [32]. Other problems to be addressed concern ultracold atoms in optical lattices subject to random potentials [33], which are promising not only from purely scientific point of view, but also for many technological applications.…”

Section: Discussionmentioning

confidence: 99%

Trajectory Statistics of Confined L\'evy Flights and Boltzmann-type Equilibria

Żaba¹,

Garbaczewski²,

Stephanovich³

2013

Acta Phys. Pol. B

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We analyze a specific class of random systems that are driven by a symmetric Lévy stable noise, where the Langevin representation is absent. In view of the Lévy noise sensitivity to environmental inhomogeneities, the pertinent random motion asymptotically sets down at the Boltzmann-type equilibrium, represented by a probability density function (pdfHere, we infer pdf ρ(x, t) based on numerical path-wise simulation of the underlying jump-type process. A priori given data are jump transition rates entering the master equation for ρ(x, t) and its target pdf ρ * (x). To simulate the above processes, we construct a suitable modification of the Gillespie algorithm, originally invented in the chemical kinetics context. We exemplified our algorithm simulating different jump-type processes and discuss the dynamics of real physical systems, where it can be useful.

show abstract

“…In those systems it is the properly tailored periodic "potential landscape" which enforces a conversion of a homogenous stochastic process (Brownian motion for reference) into the directed motion of particles at nanometer scales. It is closely related to the problem of so-called sorting in periodic potentials [33]. Other problems to be addresses concern ultracold atoms in optical lattices subject to random potentials [34], which are promising not only from purely scientific point of view, but also for many technological applications.…”

Section: Discussionmentioning

confidence: 99%

Trajectory statistics of confined Lévy flights and Boltzmann-type equilibria

Zaba,

Garbaczewski,

Stephanovich

2013

Preprint

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We analyze a specific class of random systems that are driven by a symmetric Lévy stable noise, where Langevin representation is absent. In view of the Lévy noise sensitivity to environmental inhomogeneities, the pertinent random motion asymptotically sets down at the Boltzmann-type equilibrium, represented by a probability density function (pdf) ρ * (x) ∼ exp [−Φ(x)]. Here, we infer pdf ρ(x, t) based on numerical path-wise simulation of the underlying jump-type process. A priori given data are jump transition rates entering the master equation for ρ(x, t) and its target pdf ρ * (x). To simulate the above processes, we construct a suitable modification of the Gillespie algorithm, originally invented in the chemical kinetics context. We exemplified our algorithm simulating different jump-type processes and discuss the dynamics of real physical systems where it can be useful. I. INTRODUCTIONDespite many attempts to pin down the essential features of dynamics and relaxation in random systems, the problem is still far from its complete solution. It turns out that Lévy flight models are adequate for the description of different random systems ranging from the motion of defects in disordered solids to the dynamics of assets in stock markets, see, e.g. [1]. They are especially useful to model the random systems on the semi-phenomenological, mesoscopic level, when the (often unknown) details of their microscopic random behavior are substituted by a properly tailored (e.g. based on experimental data) (Gaussian or Lévy) noise. Paradoxically, in disordered solids, the noise can promote order and organization, switching them between different equilibrium states. The latter situation emerges in disordered ferroelectrics, where the fluctuations of order parameter (spontaneous polarization) give rise to selflocalization of charge carrier, generating a fluctuon, an analog of well-known polaron in disordered substance [2]. These fluctuons make a substantial contribution into conductivity and optical properties of disordered dielectrics.Many random processes admit a description based on stochastic differential equations. In such case there is a routine passage procedure from microscopic random variables to macroscopic (statistical ensemble) data. The latter are encoded in the time evolution of an associated probability density function (pdf) which is a solution of a deterministic transport equation. A paradigmatic example is the so-called Langevin modeling of diffusion-type and jump-type processes. The presumed microscopic model of the dynamics in external force fields is provided by the Langevin (stochastic) equation whose direct consequence is the Fokker-Planck equation, [3] and [4]. We note that in case of jump-type processes the familiar Laplacian (Wiener noise generator) needs to be replaced by a suitable pseudo-differential operator (fractional Laplacian, in case of a symmetric Lévy-stable noise).We pay a particular attention to jump-type processes which are omnipresent in Nature (see [5] and references therein). Their chara...

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Directed transport in periodically rocked random sawtooth potentials

Cited by 9 publications

References 26 publications

Lévy flights in confining environments: Random paths and their statistics

Lévy flights in confining environments: Random paths and their statistics

Trajectory Statistics of Confined L\'evy Flights and Boltzmann-type Equilibria

Trajectory statistics of confined Lévy flights and Boltzmann-type equilibria

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