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2019 | Buch

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Statistical Mechanics of Classical and Disordered Systems

Luminy, France, August 2018

herausgegeben von: Prof. Véronique Gayrard, Dr. Louis-Pierre Arguin, Dr. Nicola Kistler, Dr. Irina Kourkova

Verlag: Springer International Publishing

Buchreihe : Springer Proceedings in Mathematics & Statistics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

These proceedings of the conference Advances in Statistical Mechanics, held in Marseille, France, August 2018, focus on fundamental issues of equilibrium and non-equilibrium dynamics for classical mechanical systems, as well as on open problems in statistical mechanics related to probability, mathematical physics, computer science, and biology.

Statistical mechanics, as envisioned more than a century ago by Boltzmann, Maxwell and Gibbs, has recently undergone stunning twists and developments which have turned this old discipline into one of the most active areas of truly interdisciplinary and cutting-edge research.

The contributions to this volume, with their rather unique blend of rigorous mathematics and applications, outline the state-of-the-art of this success story in key subject areas of equilibrium and non-equilibrium classical and quantum statistical mechanics of both disordered and non-disordered systems.

Aimed at researchers in the broad field of applied modern probability theory, this book, and in particular the review articles, will also be of interest to graduate students looking for a gentle introduction to active topics of current research.

Inhaltsverzeichnis

Frontmatter

Ordered Systems

Frontmatter

Gibbs-Non Gibbs Transitions in Different Geometries: The Widom-Rowlinson Model Under Stochastic Spin-Flip Dynamics

Abstract

The Widom-Rowlinson model is an equilibrium model for point particles in Euclidean space. It has a repulsive interaction between particles of different colors, and shows a phase transition at high intensity. Natural versions of the model can moreover be formulated in different geometries: in particular as a lattice system or a mean-field system. We will discuss recent results on dynamical Gibbs-non Gibbs transitions in this context. Main issues will be the possibility or impossibility of an immediate loss of the Gibbs property, and of full-measure discontinuities of the time-evolved models.

Christof Külske

One-Sided Versus Two-Sided Stochastic Descriptions

Abstract

It is well-known that discrete-time finite-state Markov Chains, which are described by one-sided conditional probabilities which describe a dependence on the past as only dependent on the present, can also be described as one-dimensional Markov Fields, that is, nearest-neighbor Gibbs measures for finite-spin models, which are described by two-sided conditional probabilities. In such Markov Fields the time interpretation of past and future is being replaced by the space interpretation of an interior volume, surrounded by an exterior to the left and to the right. If we relax the Markov requirement to weak dependence, that is, continuous dependence, either on the past (generalising the Markov-Chain description) or on the external configuration (generalising the Markov-Field description), it turns out this equivalence breaks down, and neither class contains the other. In one direction this result has been known for a few years, in the opposite direction a counterexample was found recently. Our counterexample is based on the phenomenon of entropic repulsion in long-range Ising (or “Dyson”) models.

Aernout C. D. van Enter

Disordered Systems

Frontmatter

The Free Energy of the GREM with Random Magnetic Field

Abstract

We study the extreme value statistics of the two-level Generalized Random Energy Model (GREM) coupled with a random magnetic field. This model generalizes both the Random Energy Model (REM) combined with a random magnetic field studied by de Oliveira Filho et al. and Arguin and Kistler, and that of the GREM in the presence of a uniform external field studied by Bovier and Klimovsky. The extreme value statistics of the model are determined at the level of the entropy and of the free energy, generalizing the behavior found in Arguin and Kistler and Bovier and Klimovsky. The proofs rely on an application of large deviation theory which reduces the problem to that of a GREM on a subset of configurations for which the magnetization attains a specific value. The methods introduced are general and extend to the k-level GREM in the presence of a random magnetic field.

Louis-Pierre Arguin, Roberto Persechino

A Morita Type Proof of the Replica-Symmetric Formula for SK

Abstract

We give a proof of the replica symmetric formula for the free energy of the Sherrington-Kirkpatrick model in high temperature which is based on the TAP formula. This is achieved by showing that the conditional annealed free energy equals the quenched one, where the conditioning is given by an appropriate \(\sigma \)-field with respect to which the TAP solutions are measurable.

Erwin Bolthausen

Concentration of the Clock Process Normalisation for the Metropolis Dynamics of the REM

Abstract

In Černý and Wassmer (Probab. Theory Relat. Fields 167:253–303, 2017) [8], it was shown that the clock process associated with the Metropolis dynamics of the Random Energy Model converges to an \(\alpha \)-stable process, after being scaled by a random, Hamiltonian dependent, normalisation. We prove here that this random normalisation can be replaced by a deterministic one.

Jiří Černý

Dynamic Phase Diagram of the REM

Abstract

By studying the two-time overlap correlation function, we give a comprehensive analysis of the phase diagram of the Random Hopping Dynamics of the Random Energy Model (REM) on time-scales that are exponential in the volume. These results are derived from the convergence properties of the clock process associated to the dynamics and fine properties of the simple random walk in the n-dimensional discrete cube.

Véronique Gayrard, Lisa Hartung

The Replica Trick in the Frame of Replica Interpolation

Abstract

As it is very well known, for disordered models of statistical mechanics, the celebrated “replica trick” is based on the idea that the annealed averages for replicated systems give some relevant information on the original system. We give a new interpretation of the replica trick in the general frame of interpolation on the number of replicas, extending on the customary exploitation of the replica trick as connected with analytic continuation toward zero replicas. The case of the Derrida Random Energy Model is synthetically worked out in the frame of the replica interpolation. We give also some application concerning the so called Almeida-Thouless line in the Sherrington-Kirkpatrick mean field spin glass model.

Francesco Guerra

From Parisi to Boltzmann

Gibbs Potentials and High Temperature Expansions in Mean Field

Abstract

We sketch a new framework for the analysis of disordered systems, in particular mean field spin glasses, which is variational in nature and within the formalism of classical thermodynamics. For concreteness, only the Sherrington–Kirkpatrick model is considered here. For this we show how the Parisi solution (replica symmetric, or when replica symmetry is broken) emerges, in large but finite volumes, from a high temperature expansion to second order of the Gibbs potential with respect to order parameters encoding the law of the effective fields. In contrast with classical systems where convexity in the order parameters is the default situation, the functionals employed here are, at infinite temperature, concave: this feature is eventually due to the Gaussian nature of the interaction and implies, in particular, that the canonical Boltzmann-Gibbs variational principles must be reversed. The considerations suggest that thermodynamical phase transitions are intimately related to the divergence of the infinite expansions.

Goetz Kersting, Nicola Kistler, Adrien Schertzer, Marius A. Schmidt

Nature Versus Nurture: Dynamical Evolution in Disordered Ising Ferromagnets

Abstract

We study the predictability of zero-temperature Glauber dynamics in various models of disordered ferromagnets. This is analyzed using two independent dynamical realizations with the same random initialization (called twins). We derive, theoretically and numerically, trajectories for the evolution of the normalized magnetization and twin overlap as the system size tends to infinity. The systems we treat include mean-field ferromagnets with light-tailed and heavy-tailed coupling distributions, as well as highly-disordered models with a variety of other geometries. In the mean-field setting with light-tailed couplings, the disorder averages out and the limiting trajectories of the magnetization and twin overlap match those of the homogenous Curie–Weiss model. On the other hand, when the coupling distribution has heavy tails, or the geometry changes, the effect of the disorder persists in the thermodynamic limit. Nonetheless, qualitatively all such random ferromagnets share a similar time evolution for their twin overlap, wherein the two twins initially decorrelate, before either partially or fully converging back together due to the ferromagnetic drift.

Lily Z. Wang, Reza Gheissari, Charles M. Newman, Daniel L. Stein

Miscellaneous

Frontmatter

Tightness and Line Ensembles for Brownian Polymers Under Geometric Area Tilts

Abstract

We prove tightness and limiting Brownian-Gibbs description for line ensembles of non-colliding Brownian bridges above a hard wall, which are subject to geometrically growing self-potentials of tilted area type. Statistical properties of the resulting ensemble are very different from that of non-colliding Brownian bridges without self-potentials. The model itself was introduced in order to mimic level lines of \(2+1\) discrete Solid-On-Solid random interfaces above a hard wall.

Pietro Caputo, Dmitry Ioffe, Vitali Wachtel

Large Deviations and Uncertainty Relations in Periodically Driven Markov Chains

Abstract

We present some of the results contained in [2, 3, 6], giving in a synthetic form a flavor of the questions faced there. In particular, we present large deviation principles for the extended empirical measure, flow and current of Markov chains with time-periodic jump rates. As an application we derive a Gallavotti–Cohen duality relation for the fluctuating entropy flow and we also derive trade-off relations between speed and precision (called generalized thermodynamic uncertainty relations) for a broad class of functionals of stochastic trajectories. These theoretical results find applications in the thermodynamics of small systems, as biomolecular motors and molecular pumps.

Alessandra Faggionato

Titel: Statistical Mechanics of Classical and Disordered Systems
herausgegeben von: Prof. Véronique Gayrard
Dr. Louis-Pierre Arguin
Dr. Nicola Kistler
Dr. Irina Kourkova
Verlag: Springer International Publishing
Electronic ISBN: 978-3-030-29077-1
Print ISBN: 978-3-030-29076-4
DOI: https://doi.org/10.1007/978-3-030-29077-1