Large deviations from the hydrodynamical limit for a system of independent brownian particles

Kipnis, Claude; Olla, Stefano

doi:10.1080/17442509008833661

Cited by 35 publications

(47 citation statements)

References 2 publications

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“…In this spirit the large deviations from the hydrodynamical limit for independent Brownian motions were studied in [7], using an approach that can be also used for strongly interacting systems, like simple exclusion, as we show in this paper.…”

Section: Introductionmentioning

confidence: 99%

Hydrodynamics and large deviation for simple exclusion processes

Kipnis

Olla

Varadhan

1989

Comm Pure Appl Math

253

296

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We prove the hydrodynamical limit for weakly asymmetric simple exclusion processes. A large deviation property with respect to this limit is established for the symmetric case. We treat also the situation where a slow reaction (creation and annihilation of particles) is present. IntroductionWhen we study the hydrodynamical limit (reduced description) of an infinite particle system we can obtain the same limiting equation for different systems. For instance, both symmetric simple exclusion and a system of independent symmetric random walks give rise to the heat equation. However, in the study of the large deviations, the rate functions are different for these two systems.In this spirit the large deviations from the hydrodynamical limit for independent Brownian motions were studied in [7], using an approach that can be also used for strongly interacting systems, like simple exclusion, as we show in this paper.The case of independent particles is simple because only the empirical density field appears in the martingales considered. In the case of strongly interacting systems the density field is no longer autonomous and we need to control the deviations of the empirical correlation fields. In Theorem 2.1 we prove that the probability that any correlation field deviates from an appropriate function of the density field is superexponentially small, whereas the large deviations of the density fields are only exponentially small.The other ingredient needed for the lower bound is a large number of hydrodynamical limit theorems for several small perturbations of the original

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Section: Introductionmentioning

confidence: 99%

Hydrodynamics and large deviation for simple exclusion processes

Kipnis

Olla

Varadhan

1989

Comm Pure Appl Math

253

296

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“…One can also consider models that allow more than one particle at each site, such as the zero-range process (see, e.g., [14,167]), or models with particle reservoirs that add and remove particles at given rates. Moreover, one can choose not to impose the exclusion rule, in which case the particles jump independently of one another [159,160].…”

Section: Interacting Particle Modelsmentioning

confidence: 99%

The large deviation approach to statistical mechanics

2009

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The theory of large deviations is concerned with the exponential decay of probabilities of large fluctuations in random systems. These probabilities are important in many fields of study, including statistics, finance, and engineering, as they often yield valuable information about the large fluctuations of a random system around its most probable state or trajectory. In the context of equilibrium statistical mechanics, the theory of large deviations provides exponential-order estimates of probabilities that refine and generalize Einstein's theory of fluctuations. This review explores this and other connections between large deviation theory and statistical mechanics, in an effort to show that the mathematical language of statistical mechanics is the language of large deviation theory. The first part of the review presents the basics of large deviation theory, and works out many of its classical applications related to sums of random variables and Markov processes. The second part goes through many problems and results of statistical mechanics, and shows how these can be formulated and derived within the context of large deviation theory. The problems and results treated cover a wide range of physical systems, including equilibrium many-particle systems, noise-perturbed dynamics, nonequilibrium systems, as well as multifractals, disordered systems, and chaotic systems. This review also covers many fundamental aspects of statistical mechanics, such as the derivation of variational principles characterizing equilibrium and nonequilibrium states, the breaking of the Legendre transform for nonconcave entropies, and the characterization of nonequilibrium fluctuations through fluctuation relations.PACS numbers: 65.40.Gr, Typeset by REVT E X arXiv:0804.0327v2 [cond-mat.stat-mech]

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“…Large deviations for a rather general class of locally mean-field type models were investigated in [14] via a careful adaptation of ideas and techniques which were originally introduced by Dawson and Gärtner [5,4]. It seems, however, that in a particular case we consider here, our results on the existence and uniqueness of strong solutions to the system (L-MV) and, accordingly, on hydrodynamic limits towards these strong solutions, pave the way to for a simpler and more transparent proof of the large deviation principle for the law P N T of the empirical measure µ N = µ N [0, T ] on C [0, T ], M 1 (R × T d ) , which relies on martingale techniques of [3,12], see also Section 4.2.1 of [8] for a very clear exposition of the method. Below we sketch the corresponding argument.…”

Section: Large Deviationsmentioning

confidence: 99%