Stationarization and Multithermalization in spin glasses

Abstract

We develop further the study of a system in contact with a multibath having different temperatures at widely separated timescales. We consider those systems that do not thermalize in finite times when in contact with an ordinary bath but may do so in contact with a multibath. Thermodynamic integration is possible, thus allowing one to recover the stationary distribution on the basis of measurements performed in a `multi-reversible' transformation. We show that following such a protocol the system is at each step described by a generalization of the Boltzmann-Gibbs distribution, that has been studied in the past. Guerra's bound interpolation scheme for spin-glasses is closely related to this: by translating it into a dynamical setting, we show how it may actually be implemented in practice. The phase diagram plane of temperature vs ``number of replicas", long studied in spin- glasses, in our approach becomes simply that of the two temperatures the system is in contact with. We suggest that this representation may be used to directly compare phenomenological and mean-field inspired models. Finally, we show how an approximate out of equilibrium probability distribution may be inferred experimentally on the basis of measurements along an almost reversible transformation.

Stationary non-equilibrium measure for a dynamics with two temperatures and two widely different time scales

“…We discuss a multibath spin-glass model with fast soft spins and slow external magnetic fields. This kind of model has been discussed recently in [17,18] where it is argued on heuristic grounds that Guerra's hierarchical measure for Replica Symmetry Breaking (RSB) appears as the stationary measure of the multibath Langevin dynamics. We show that our analysis is applicable for a finite system.…”

“…The heuristic argument above can be generalized to systems where more than two families of degrees of freedom have widely separated time-scales and different temperatures [18] and the analogous of expression ( 5) is obtained through a hierarchy of conditional averages. As it turns out, the resulting measure is at the core of Guerra's interpolation scheme [19] which leads to the Talagrand's proof [20] of the celebrated Parisi solution [21,22] for the Sherringhton-Kirkpatrick model.…”

“…We also mention that the hierarchical structure of the measure, as well the Parisi solution itself, can be represented in terms of Ruelle Probability Cascades [23,24], a central object in the mathematical theory of replica symmetry breaking in mean-field spin glasses [25,26]. Some aspects of the connection between multibath models and replica symmetry breaking has been recently investigated in [18].…”

“…showing that our estimates give the correct time scales for approaching the stationary distribution as well as the correct order of magnitude O(λ −1 ) for the residual KL divergences. The second example in Subsection 3.2 is a spin glass model where the spin degrees of freedom as well as the external magnetic fields are given a dynamics in the spirit of [17,18]. The third example (Subsection 3.3) comes from a paradigm in modern high-dimensional inference and estimation theory, namely the problem of estimating a rank-one matrix which is observed through a noisy channel (see [31] for a general review on the subject and a discussion of applications).…”

Time-reparametrization invariances, multithermalization and the Parisi scheme

“…Different temperatures are possible, but in diferent 'scales', a notion one has to define. We refer to this situation as 'multithermalization' [11,12]. Rather unexpectedly, the temperatures involved in the slow dynamics coincide with a series of parameters computed for equilibrium in the Parisi scheme.…”