[Paper Review] The relation between Parisi scheme and multi-thermalized dynamics in finite dimensions

Silvio Franz, Jorge Kurchan|arXiv (Cornell University)|Jul 12, 2022

Theoretical and Computational Physics4 citations

TL;DR

This paper establishes a one-to-one correspondence between the ultrametric Parisi replica scheme and multithermalized out-of-equilibrium dynamics in finite-dimensional spin glasses. It shows that stochastic stability of equilibrium states implies multithermalization in dynamics, and vice versa, under time-reparametrization invariance, linking static RSB to slow, aging dynamics via linear response theory and susceptibility scaling.

ABSTRACT

In this note we summarize the connections between equilibrium and slow out of equilibrium dynamics in finite dimensional glasses, such as we understand them today. If we assume that a finite-dimensional system is stable with respect to a family of weak random perturbations (stochastic stability), then its dynamics have a `Multithermalization' structure if and only if the Boltzmann-Gibbs distribution obeys an Ultrametric Parisi distribution.

Motivation & Objective

• To clarify the deep connection between equilibrium replica symmetry breaking (RSB) and slow out-of-equilibrium dynamics in finite-dimensional spin glasses.
• To investigate whether the ultrametric Parisi scheme and multithermalization structure are equivalent under stochastic stability and reparametrization invariance.
• To assess the validity of the one-step RSB and full RSB scenarios in finite dimensions using dynamic and equilibrium response functions.
• To explore whether multithermalization with multiple effective temperatures emerges naturally from the Parisi solution under weak random perturbations.
• To reconcile mean-field predictions with finite-dimensional behavior through linear response and susceptibility analysis

Proposed method

• Uses linear response theory to relate dynamical response functions to equilibrium correlation functions in the slow-dynamics regime.
• Applies the method of random perturbations to test stochastic stability of the equilibrium measure and derive constraints on overlap statistics.
• Employs the Ghirlanda-Guerra identities as a condition to enforce ultrametricity in the overlap distribution, linking them to the Parisi RSB solution.
• Analyzes time-reparametrization invariance as a key symmetry ensuring consistency between equilibrium and slow dynamics.
• Compares numerical and experimental data on correlation decay and susceptibility in 3D and 4D systems to test for multithermalization and dynamic ultrametricity.
• Uses the FMPP strategy (Franz-Mézard-Parisi-Peliti) to equate bulk dynamical quantities with equilibrium ones under slow relaxation

Experimental results

Research questions

• RQ1 Does the ultrametric Parisi RSB scheme imply multithermalized dynamics in finite-dimensional systems under stochastic stability?
• RQ2 Is multithermalization in dynamics equivalent to the existence of a full RSB equilibrium state in finite dimensions?
• RQ3 Can time-reparametrization invariance be derived from the Parisi scheme, and does it underlie the consistency between equilibrium and slow dynamics?
• RQ4 Why do experimental and numerical studies in 3D show only two timescales despite expectations of multiple effective temperatures?
• RQ5 Is one-step RSB dynamically unstable in finite dimensions, favoring full RSB and ultrametricity?

Key findings

• The Parisi RSB scheme and multithermalized dynamics are equivalent if and only if the system is stochastically stable and exhibits time-reparametrization invariance.
• Stochastic stability of the equilibrium measure implies that the overlap distribution must be ultrametric, enforcing the full RSB structure.
• In 4D systems, numerical evidence supports dynamic ultrametricity with multiple timescales, consistent with the Sherrington-Kirkpatrick model.
• In 3D, experimental data showing only two timescales may reflect preasymptotic effects, not a violation of multithermalization.
• The one-step RSB state is likely dynamically unstable in finite dimensions, favoring full RSB and ultrametricity due to marginal stability against continuous RSB.
• Linear response theory allows exact mapping of dynamical response to equilibrium correlations under slow relaxation, provided reparametrization invariance holds