Exact solutions of some urn models of relaxation in glassy dynamics

Arora, D.; Bhatia, D. P.; Prasad, Mahabir

doi:10.1103/physreve.60.145

Cited by 4 publications

(9 citation statements)

References 6 publications

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“…∼ k log k − k, giving a logarithmic correction on a semi-log plot. For larger values of k, a different approach is needed; however, Arora et al [19] showed that for k = N the asymptotic result is τ FP N ∼ V N , which coincides with the asymptotic behavior of our result when we put V = N/ρ. Comparison between mean-field and spatial cases.…”

supporting

confidence: 88%

“…Since diffusive processes give rise to spatial and temporal correlations that render the problem analytically intractable, we also consider a mean-field version of this problem, in which particles can jump not only between neighbouring sites, but to any of the sites in the system with equal probability, which reduces the problem to a type of Ehrenfest urn model [13,19]. We show numerically that this approximation provides a qualitative explanation of the behaviour of the mean first-passage time also for diffusive dynamics.…”

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confidence: 99%

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How rare are diffusive rare events?

Sanders

Larralde

2008

Europhys. Lett.

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We study the time until first occurrence, the first-passage time, of rare density fluctuations in diffusive systems. We approach the problem using a model consisting of many independent random walkers on a lattice. The existence of spatial correlations makes this problem analytically intractable. However, for a mean-field approximation in which the walkers can jump anywhere in the system, we obtain a simple asymptotic form for the mean first-passage time to have a given number k of particles at a distinguished site. We show numerically, and argue heuristically, that for large enough k, the mean-field results give a good approximation for firstpassage times for systems with nearest-neighbour dynamics, especially for two and higher spatial dimensions. Finally, we show how the results change when density fluctuations anywhere in the system, rather than at a specific distinguished site, are considered.

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confidence: 88%

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confidence: 99%

How rare are diffusive rare events?

Sanders

Larralde

2008

Europhys. Lett.

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“…Relaxation times to reach the ground state at T = 0 do of course diverge with the system size (as 2 N for the Backgammon model [156,157,161], or more slowly as N 2 [161] in variants such as model C from [151]), but at T > 0 the final energy per box or particle lies above the ground state by a finite amount and so all timescales remain finite.…”

Section: Relaxation Timescales and Dynamical Transitionsmentioning

confidence: 99%

Glassy dynamics of kinetically constrained models

Ritort

Sollich

2003

Advances in Physics

751

1,055

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We review the use of kinetically constrained models (KCMs) for the study of dynamics in glassy systems. The characteristic feature of KCMs is that they have trivial, often non-interacting, equilibrium behaviour but interesting slow dynamics due to restrictions on the allowed transitions between configurations. The basic question which KCMs ask is therefore how much glassy physics can be understood without an underlying "equilibrium glass transition". After a brief review of glassy phenomenology, we describe the main model classes, which include spin-facilitated (Ising) models, constrained lattice gases, models inspired by cellular structures such as soap froths, models obtained via mappings from interacting systems without constraints, and finally related models such as urn, oscillator, tiling and needle models. We then describe the broad range of techniques that have been applied to KCMs, including exact solutions, adiabatic approximations, projection and mode-coupling techniques, diagrammatic approaches and mappings to quantum systems or effective models. Finally, we give a survey of the known results for the dynamics of KCMs both in and out of equilibrium, including topics such as relaxation time divergences and dynamical transitions, nonlinear relaxation, aging and effective temperatures, cooperativity and dynamical heterogeneities, and finally non-equilibrium stationary states generated by external driving. We conclude with a discussion of open questions and possibilities for future work.

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“…Let us consider a system quenched down to T = 0 and the aging regime reached in the asymptotic long-time regime where (dE/dt)/E << 1. In that limit, the system has a number of empty boxes N empty = −E and further decrease of that number by one unit ∆E = −1 requires a time that exceedingly grows as E decreases toward its minimum value E = −N + 1 (all particles condensed into a single box) [197,198]. Therefore, as relaxation slows down, for a long time the system wanders through the constant energy surface by exchanging particles among occupied boxes.…”

Section: The Backgammon and Urn Modelsmentioning

confidence: 99%

Violation of the fluctuation–dissipation theorem in glassy systems: basic notions and the numerical evidence

Crisanti

Ritort

2003

J. Phys. A: Math. Gen.

359

595

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Abstract. This review reports on the research done during the past years on violations of the fluctuation-dissipation theorem (FDT) in glassy systems. It is focused on the existence of a quasi-fluctuation-dissipation theorem (QFDT) in glassy systems and the currently supporting knowledge gained from numerical simulation studies.It covers a broad range of non-stationary aging and stationary driven systems such as structural-glasses, spin-glasses, coarsening systems, ferromagnetic models at criticality, trap models, models with entropy barriers, kinetically constrained models, sheared systems and granular media. The review is divided into four main parts: 1) An introductory section explaining basic notions related to the existence of the FDT in equilibrium and its possible extension to the glassy regime (QFDT), 2) A description of the basic analytical tools and results derived in the framework of some exactly solvable models, 3) A detailed report of the current evidence in favour of the QFDT and 4) A brief digression on the experimental evidence in its favour. This review is intended for inexpert readers who want to learn about the basic notions and concepts related to the existence of the QFDT as well as for the more expert readers who may be interested in more specific results.

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Exact solutions of some urn models of relaxation in glassy dynamics

Cited by 4 publications

References 6 publications

How rare are diffusive rare events?

How rare are diffusive rare events?

Glassy dynamics of kinetically constrained models

Violation of the fluctuation–dissipation theorem in glassy systems: basic notions and the numerical evidence

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