Report on 2101.12702v3

The Parisi scheme for equilibrium and the corresponding slow dynamics with multithermalization -same temperature common to all observables, different temperatures only possible at widely separated timescales -imply one another.Consistency requires that two systems brought into infinitesimal coupling be able to rearrange their timescales in order that all their temperatures match: this time reorganisation is only possible because the systems have a set of time-reparametrization invariances, that are thus seen to be an essential component of the scenario.CONTENTS I. Introduction 1 A. Equilibrium 2 B. Dynamics 2 II.The framework 4 A. Factoring out time -general kinematic constraints.4 B. Three examples 4 C. Time-reparametrizations 5 D. Dynamic multithermalization properties 6 III.Connections between dynamic and Parisi scheme 7 A. A first, formal bridge between dynamic and static (replica) and calculations 7 IV.Properties derived from stochastic stability 8 A. Same temperatures for all observables implies separation of timescales 8 B. Relation between statics and dynamics in finite dimensions 9 V.The role of reparametrization invariance(s) 14 A. How do the families of reparametrization invariances come about 14 B. Two glasses and a wormhole 15 VI.Conclusion 17 References 17 I. INTRODUCTIONA finite dimensional system whose equilibrium solution follows the Parisi scheme [1] will take an infinite time to reach this equilibrium starting form random configuration.It may also be driven into an out of equilibrium steadystate by an infinitesimal drive, such as shear [2, 3], or time-dependence of disorder [4].If the relaxation times are long, or, in a steady-state, if the drive is weak, the dynamics are slow: this is the regime we are interested in.The idea of this paper is composed of two parts:• The out of equilibrium dynamics under these circumstances is a very specific one [4][5][6][7][8][9].At given times one may define a temperature with a thermodynamic meaning [10], it is the same for all observables.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