Velocity distribution for quasistable acceleration in the presence of multiplicative noise

Abstract. The diusion-based growth of islands composed of clusters of metal atoms on a substrate is considered in the aggregation regime. A stochastic approach is proposed to describe the dynamics of island growth based on a Langevin equation with multiplicative noise. The distribution of island sizes, obtained as a solution of the corresponding Fokker-Planck equation, is derived. The time-dependence of island growth on its fractal dimension is analysed. The eect of mobility of the small islands on the growth of large islands is considered. Numerical simulations are in a good agreement with theoretical results.

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“…and coincides with the distribution obtained in [73] for a = 0. Then, the expression of the mean of x(t) is …”

Section: Fokker-planck Equation For Island Size Distributionsupporting

confidence: 89%

Stochastic model for the epitaxial growth of two-dimensional islands in the submonolayer regime

et al. 2016

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“…If this is the case, then one can consider a continuous driving protocol instead of repeated quenches and all the results will be the same. Such a setup was analyzed by Jarzynski [213] followed by other works [193,[220][221][222][223][224]]. An interesting and nontrivial result that emerges from this analysis is a nonequilibrium exponential velocity distribution (to be contrasted with the Gaussian Maxwell distribution).…”

Section: Heating a Particle In A Fluctuating Chaotic Cavitymentioning

confidence: 99%

From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics

D’Alessio

Kafri

Polkovnikov

et al. 2016

Advances in Physics

2,144

2,527

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This review gives a pedagogical introduction to the eigenstate thermalization hypothesis (ETH), its basis, and its implications to statistical mechanics and thermodynamics. In the first part, ETH is introduced as a natural extension of ideas from quantum chaos and random matrix theory. To this end, we present a brief overview of classical and quantum chaos, as well as random matrix theory and some of its most important predictions. The latter include the statistics of energy levels, eigenstate components, and matrix elements of observables. Building on these, we introduce the ETH and show that it allows one to describe thermalization in isolated chaotic systems without invoking the notion of an external bath. We examine numerical evidence of eigenstate thermalization from studies of many-body lattice systems. We also introduce the concept of a quench as a means of taking isolated systems out of equilibrium, and discuss results of numerical experiments on quantum quenches. The second part of the review explores the implications of quantum chaos and ETH to thermodynamics. Basic thermodynamic relations are derived, including the second law of thermodynamics, the fundamental thermodynamic relation, fluctuation theorems, the fluctuation-dissipation relation, and the Einstein and Onsager relations. In particular, it is shown that quantum chaos allows one to prove these relations for individual Hamiltonian eigenstates and thus extend them to arbitrary stationary statistical ensembles. In some cases, it is possible to extend their regimes of applicability beyond the standard thermal equilibrium domain. We then show how one can use these relations to obtain nontrivial universal energy distributions in continuously driven systems. At the end of the review, we briefly discuss the relaxation dynamics and description after relaxation of integrable quantum systems, for which ETH is violated. We present results from numerical experiments and analytical studies of quantum quenches at integrability. We introduce the concept of the generalized Gibbs ensemble, and discuss its connection with ideas of prethermalization in weakly interacting systems.

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“…The assumption ε ≪ 1 is not particularly restrictive; even if particles begin with an initial speed comparable to or less than u c , energy gaining collisions are more likely than energy losing collisions due to the flux factor in the biased distribution, and a slow particle will gain roughly mu 2 c of energy during a collision according to Eq. (18). Therefore, a slow particle will more than likely gain speed u c ∼ O(1) during a single bounce, and after 1/δ bounces, where δ ≪ 1 is some small number, the particle will more than likely have a speed v such that u c /v δ ≪ 1.…”

Section: Energy Statisticsmentioning

confidence: 99%

“…The details of the separation of variables, including existence, uniqueness, and boundary conditions, are given in Ref. [18] and will be omitted here. We also acknowledge a similar, much older, one-dimensional solution given in Ref.…”

Section: Energy Diffusionmentioning

confidence: 99%

Universal energy diffusion in a quivering billiard

Demers

Jarzynski

2015

Phys. Rev. E

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We introduce and study a model of time-dependent billiard systems with billiard boundaries undergoing infinitesimal wiggling motions. The so-called quivering billiard is simple to simulate, straightforward to analyze, and is a faithful representation of time-dependent billiards in the limit of small boundary displacements. We assert that when a billiard's wall motion approaches the quivering motion, deterministic particle dynamics become inherently stochastic. Particle ensembles in a quivering billiard are shown to evolve to a universal energy distribution through an energy diffusion process, regardless of the billiard's shape or dimensionality, and as a consequence universally display Fermi acceleration. Our model resolves a known discrepancy between the one-dimensional Fermi-Ulam model and the simplified static wall approximation. We argue that the quivering limit is the true fixed wall limit of the Fermi-Ulam model.

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Velocity distribution for quasistable acceleration in the presence of multiplicative noise

Cited by 13 publications

References 25 publications

Stochastic model for the epitaxial growth of two-dimensional islands in the submonolayer regime

Stochastic model for the epitaxial growth of two-dimensional islands in the submonolayer regime

From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics

Universal energy diffusion in a quivering billiard

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