[Paper Review] Memories from the ergodic phase: the awkward dynamics of spherical mixed p-spin models

Giampaolo Folena, Silvio Franz+1 more|arXiv (Cornell University)|Mar 4, 2019

Theoretical and Computational Physics4 citations

TL;DR

This paper investigates the out-of-equilibrium dynamics of spherical mixed p-spin models after quenching from the ergodic phase. Using numerical integration of dynamical mean-field equations, it identifies an unexpected dynamical phase transition below an onset temperature $T_{\text{onset}} > T_{\text{MCT}}$, where memory of the initial condition persists due to relaxation toward deep marginal minima, resembling glass-forming liquids but defying simple aging descriptions.

ABSTRACT

We revisit the long-time limit of the out of equilibrium dynamics of mean-field spherical mixed p-spin models. We consider quenches (gradient descent dynamics) starting from initial conditions thermalized at some temperature in the ergodic phase. We perform numerical integration of the dynamical mean-field equations of the model and we find an unexpected dynamical phase transition. Below an onset temperature $T_{onset}$, higher than the dynamical transition temperature $T_{MCT} $, the asymptotic goes below the threshold energy of the dominant marginal minima of the function and memory of the initial condition is kept. This behavior, not present in the pure spherical p-spin , resembles closely the one observed in simulations of glass forming liquids. We then investigate the nature of asymptotic dynamics, finding an aging state that relaxes towards deep marginal minima. Careful analysis however rules out simple aging solutions, leaving open a full comprehension of the memory effect in these models.

Motivation & Objective

• To understand the long-time behavior of spherical mixed p-spin models after quenching from the ergodic phase.
• To investigate whether memory of initial thermal conditions persists in the asymptotic dynamics.
• To determine the nature of the asymptotic state, particularly whether it exhibits aging or other non-equilibrium features.
• To clarify the origin of memory effects in models that differ from the pure spherical p-spin model.

Proposed method

• Numerical integration of the dynamical mean-field equations derived for spherical mixed p-spin models.
• Simulation of gradient descent dynamics starting from initial conditions thermalized at finite temperature in the ergodic phase.
• Analysis of the asymptotic energy evolution to detect memory effects and phase transitions.
• Comparison of the dynamics with the pure spherical p-spin model to isolate the role of mixed p-spin interactions.
• Identification of the threshold energy of dominant marginal minima as a reference for asymptotic energy behavior.
• Use of the onset temperature $T_{\text{onset}}$ as a critical parameter to distinguish dynamical regimes.

Experimental results

Research questions

• RQ1 Does the asymptotic dynamics of spherical mixed p-spin models retain memory of the initial thermal state?
• RQ2 What is the role of $T_{\text{onset}}$ in determining the onset of memory effects in the long-time limit?
• RQ3 How does the asymptotic energy evolution compare to the threshold energy of marginal minima?
• RQ4 Is the observed memory effect consistent with simple aging dynamics?
• RQ5 Why do mixed p-spin models exhibit memory effects not seen in the pure spherical p-spin model?

Key findings

• An unexpected dynamical phase transition occurs below an onset temperature $T_{\text{onset}}$, which is higher than the MCT transition temperature $T_{\text{MCT}}$.
• Below $T_{\text{onset}}$, the asymptotic energy falls below the threshold energy of the dominant marginal minima, indicating persistent memory of the initial condition.
• The system evolves into an aging state that relaxes toward deep marginal minima, suggesting non-trivial long-time dynamics.
• The memory effect observed is incompatible with simple aging solutions, indicating a more complex dynamical structure.
• The behavior closely resembles that seen in glass-forming liquids, despite the absence of such features in the pure spherical p-spin model.