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.
Highlights
IntroductionThe properties that are relevant for them such as large viscosity and aging, are essentially dynamical in nature
We slowly change parameters of multibath, temperature and intensity γ, always verifying that the timescale of the slow bath is of the order of the α timescale, and that the effective slow temperatures of bath and system are the same. If such a procedure is possible, and we can take the system to the liquid situation in which it is in ordinary equilibrium, athermodynamic integration is legitimate, and we have in effect experimentally proven, following the results in the previous sections, that the system as it was at the ‘age’ at which we started, may be described by the multibath measure, with the amplitude we needed to ascribe to it so that the system remained stationary from the moment we connected the multibath, and was multithermalized by it
In this sense it is more akin to the general ideas of ferromagnetism from Curie to Landau, than to the Onsager solution for the Ising model. This is why we may ask if it applies to other problems, such as finite-dimensional spin glasses. The fact that it is a scheme where a symmetry group is broken into symmetry subgroups means that it is possible to propose a solution of this form in any model
Summary
The properties that are relevant for them such as large viscosity and aging, are essentially dynamical in nature. One may prove that the multi-thermalization situation is still valid, provided the system satisfies in equilibrium a Parisi scheme ( we do not have at present any model for which we may prove that this happens): we know this by extending trivially the result of Franz et al [19] From the dynamic Fluctuation-Dissipation data one may reconstruct the distribution, which is a generalization of the Boltzmann-Gibbs one Note that, for this to be the case, it is necessary that the system admits multi-thermalization at each step of the multi-reversible transformation.
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