A large-deviation view on dynamical Gibbs-non-Gibbs transitions
Aernout van Enter, Roberto Fernández, Frank den Hollander, Frank Redig
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Key words and phrases: Stochastic spin-flip dynamics, Gibbs-non-Gibbs transition, empirical measure, non-linear generator, Nisio control semigroup, large deviation principle, bad configurations, bad empirical measures, nature versus nurture.
We develop a space-time large-deviation point of view on Gibbs-non-Gibbs transitions in spin systems subject to a stochastic spin-flip dynamics. Using the general theory for large deviations of functionals of Markov processes outlined in Feng and Kurtz [11], we show that the trajectory under the spin-flip dynamics of the empirical measure of the spins in a large block in Z^d satisfies a large deviation principle in the limit as the block size tends to infinity. The associated rate function can be computed as the action functional of a Lagrangian that is the Legendre transform of a certain non-linear generator, playing a role analogous to the moment-generating function in the Gartner-Ellis theorem of large deviation theory when this is applied to finite-dimensional Markov processes. This rate function is used to define the notion of "bad empirical measures", which are the discontinuity points of the optimal trajectories (i.e., the trajectories minimizing the rate function) given the empirical measure at the end of the trajectory. The dynamical Gibbs-non-Gibbs transitions are linked to the occurrence of bad empirical measures: for short times no bad empirical measures occur, while for intermediate and large times bad empirical mea- sures are possible. A future research program is proposed to classify the various possible scenarios behind this crossover, which we refer to as a "nature-versus-nurture" transition.
We give new and explicitly computable examples of Gibbs-non-Gibbs transitions of mean-field type, using the large deviation approach introduced in [4]. These examples include Brownian motion with small variance and related diffusion processes, such as the Ornstein-Uhlenbeck process, as well as birth and death processes. We show for a large class of initial measures and diffusive dynamics both short-time conservation of Gibbsianness and dynamical Gibbs-non-Gibbs transitions.
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We consider the random point processes on a measure space X defined by the Gibbs measures associated to a given sequence of N-particle Hamiltonians H^{(N)}. Inspired by the method of Messer-Spohn for proving concentration properties for the laws of the corresponding empirical measures, we propose a number of hypotheses on H^{(N)} which are quite general, but still strong enough to extend the approach of Messer-Spohn. The hypotheses are formulated in terms of the asymptotics of the corresponding mean energy functionals. We show that in many situations the approach even yields a Large Deviation …
We establish large deviation principles (LDPs) for empirical measures associated with a sequence of Gibbs distributions on $n$-particle configurations, each of which is defined in terms of an inverse temperature $% β_n$ and an energy functional consisting of a (possibly singular) interaction potential and a (possibly weakly) confining potential. Under fairly general assumptions on the potentials, we use a common framework to establish LDPs both with speeds $β_n/n \rightarrow \infty$, in which case the rate function is expressed in terms of a functional involving the potentials, and with speed …
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SISSA（国际高等研究学院）；INFN sezione di Trieste（意大利国家核物理研究所的里雅斯特分部）；School of Mathematics, University of Bristol（布里斯托大学数学学院）；CNRS, Unité de Mathématiques Pures et Appliquées (UMPA), École Normale Supérieure de Lyon（法国国家科学研究中心，纯粹与应用数学单元（UMPA），里昂高等师范学院）；CNRS, Laboratoire de Physique (LPENS), École Normale Supérieure de Lyon（法国国家科学研究中心，物理实验室（LPENS），里昂高等师范学院）
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