Dynamics of glassy systems

TL;DR

This work surveys the slow, out-of-equilibrium dynamics of glassy systems through solvable mean-field models (notably p-spin and SK) and connects their dynamic equations to mode-coupling theory. It develops a comprehensive framework—from generating functionals and Schwinger-Dyson equations to diagrammatic MCA—capturing aging, two-time correlations, and the breakdown of fluctuation-dissipation relations, and introduces the concept of effective temperatures governing slow modes. The TAP approach and free-energy landscape analysis link dynamics to metastable state organization, yielding insights into threshold states, complexity, and the relation to Edwards’ measures and inherent structures. Together, these results articulate a coherent mean-field scenario for glassy dynamics with predictive power for real systems, while outlining critical challenges for extending to finite dimensions and incorporating dynamic heterogeneities.

Abstract

These lecture notes can be read in two ways. The first two Sections contain a review of the phenomenology of several physical systems with slow nonequilibrium dynamics. In the Conclusions we summarize the scenario derived from the solution to some solvable models (p-spin and the like) that are intimately connected to the mode coupling approach (and similar ones) to super-cooled liquids. At the end we list a number of open problems of great relevance in this context. These Sections can be read independently of the body of the paper where we present some of the basic analytic techniques used to study the out of equilibrium dynamics of classical and quantum models with and without disorder. The technical part starts wIth a brief discussion of the role played by the environment and quenched disorder in the dynamics of classical and quantum systems. Later on we expand on the dynamic functional methods and the diagrammatic expansions and resummations used to derive macroscopic equations from the microscopic dynamics. We show why the macroscopic dynamic equations for disordered models and those resulting from self-consistent approximations to non-disordered ones coincide. We review some generic properties of the slow out of equilibrium dynamics like the modifications of FDT and their link to effective temperatures, some generic scaling forms of the correlation functions, etc. Finally we solve a family of mean-field glassy models. The connection between the dynamic treatment and the analysis of the free-energy landscape is also presented. We use pedagogical examples all along these lectures to illustrate the properties and results.

Figures (21)

• Figure 1: Two snapshots of a $2d$ cut of a $3d$ lattice undergoing ferromagnetic domain growth with non-conserved order parameter after a quench from $T\to\infty$ to $T<T_c$ at time $t=0$. On the left, $t_w=10^3$ Montecarlo steps (MCs). On the right, $t_w=10^5$ MCs. Sketch of the Landau free-energy density $f(m)$ in the low-$T$ phase. The evolution of the system is partially described by the evolution of a point in this free-energy landscape. Just after the quench $m=0$ and coarsening is visualized in this plot as a static point on top of the barrier. After $t_{\sc req}$ the point falls into the well around the magnetization of the conquering domain. After $t_{\sc erg}$ ergodicity is restored and the point jumps the barrier via thermal activation clarify.
• Figure 2: Left: sketch of the viscosity against temperature approaching $T_g$. Rough comparison between different scaling forms. (Arrhenius: $A=3900$C, $\eta_0=10^{-6}$P. Vogel-Fulcher: $A=500$C, $T_0=100.$C, $\eta_0=10$P. mct: $\eta=\eta_0/(T-T_d)^\gamma$ with $T_d=300$C, $\gamma=0.7$, $\eta_0=1700$P.) Right: cooling rate dependence of the volume, $r1>r2>r3$.
• Figure 3: Characteristic times. The waiting and measuring times are experimental times$t_{\sc exp}$. The equilibration time $t_{\sc eq}$ can be shorter or longer than them leading to equilibrium or non-equilibrium dynamics, respectively.
• Figure 4: Different types of glasses: on the left, a colloidal system (image taken from Weeks); on the right a representation of the $3d$ea spin-glass model.
• Figure 5: Left: A Couette cell used to shear a liquid. The internal and external walls turn with opposite angular velocities and the fluid is included in between them. Right: a cut of the Couette cell. ( pbc: periodic boundary conditions.)