Simple Theory for the Dynamics of Mean-Field-Like Models of Glass-Forming Fluids

Dynamics and fluctuations of minimally structured glass formers

The mean-field theory (MFT) of simple structural glasses, which is exact in the limit of infinite spatial dimensions, d→∞, offers theoretical insight as well as quantitative predictions about certain features of d=3 systems. In order to more systematically relate the behavior of physical systems to MFT, however, various finite-d effects need to be accounted for. Although some efforts along this direction have already been undertaken, theoretical and technical challenges hinder progress. A general approach to sidestep many of these difficulties consists of simulating minimally structured models whose behavior smoothly converges to that described by the MFT as d increases, so as to permit a controlled dimensional extrapolation. Using this approach, we here extract the small fluctuations around the dynamical MFT captured by a standard liquid-state observable, the non-Gaussian parameter α_{2}. The results provide insight into the physical origin of these fluctuations as well as a quantitative reference with which to compare observations for more realistic glass formers.

Jamming, relaxation, and memory in a minimally structured glass former

Structural glasses form through various out-of-equilibrium processes, including temperature quenches, rapid compression (crunches), and shear. Although each of these processes should be formally understandable within the recently formulated dynamical mean-field theory (DMFT) of glasses, the numerical tools needed to solve the DMFT equations up to the relevant physical regime do not yet exist. In this context, numerical simulations of minimally structured (and therefore mean-field-like) model glass formers can aid the search for and understanding of such solutions, thanks to their ability to disentangle structural from dimensional effects. We study here the infinite-range Mari-Kurchan model under simple out-of-equilibrium processes, and we compare results with the random Lorentz gas [J. Phys. A 55, 334001 (2022)10.1088/1751-8121/ac7f06]. Because both models are mean-field-like and formally equivalent in the limit of infinite spatial dimensions, robust features are expected to appear in the DMFT as well. The comparison provides insight into temperature and density onsets, memory, as well as anomalous relaxation. This work also further enriches the algorithmic understanding of the jamming density.

Dynamics of liquids in the large-dimensional limit

In this paper we analytically derive the exact closed dynamical equations for a liquid with short-ranged interactions in large spatial dimensions using the same statistical mechanics tools employed to analyze Brownian motion. Our derivation greatly simplifies the original path-integral-based route to these equations and provides insight into the physical features associated with high-dimensional liquids and glass formation. Most importantly, our construction provides a route to the exact dynamical analysis of important related dynamical problems, as well as a means to devise cluster generalizations of the exact solution in infinite dimensions. This latter fact opens the door to the construction of increasingly accurate theories of vitrification in three-dimensional liquids.

Active matter in infinite dimensions: Fokker-Planck equation and dynamical mean-field theory at low density

We investigate the behavior of self-propelled particles in infinite space dimensions by comparing two powerful approaches in many-body dynamics: the Fokker-Planck equation and dynamical mean-field theory. The dynamics of the particles at low densities and infinite persistence time is solved in the steady state with both methods, thereby proving the consistency of the two approaches in a paradigmatic out-of-equilibrium system. We obtain the analytic expression for the pair distribution function and the effective self-propulsion to first-order in the density, confirming the results obtained in a previous paper [T. Arnoulx de Pirey et al., Phys. Rev. Lett. 123, 260602 (2019)] and extending them to the case of a non-monotonous interaction potential. Furthermore, we obtain the transient behavior of active hard spheres when relaxing from the equilibrium to the nonequilibrium steady state. Our results show how collective dynamics is affected by interactions to first-order in the density and point out future directions for further analytical and numerical solutions of this problem.

Mean-Field Caging in a Random Lorentz Gas

The random Lorentz gas (RLG) is a minimal model of both percolation and glassiness, which leads to a paradox in the infinite-dimensional, d → ∞ limit: the localization transition is then expected to be continuous for the former and discontinuous for the latter. As a putative resolution, we have recently suggested that, as d increases, the behavior of the RLG converges to the glassy description and that percolation physics is recovered thanks to finite-d perturbative and nonperturbative (instantonic) corrections [Biroli et al. Phys. Rev. E 2021, 103, L030104]. Here, we expand on the d → ∞ physics by considering a simpler static solution as well as the dynamical solution of the RLG. Comparing the 1/d correction of this solution with numerical results reveals that even perturbative corrections fall out of reach of existing theoretical descriptions. Comparing the dynamical solution with the mode-coupling theory (MCT) results further reveals that, although key quantitative features of MCT are far off the mark, it does properly capture the discontinuous nature of the d → ∞ RLG. These insights help chart a path toward a complete description of finite-dimensional glasses.

A direct link between active matter and sheared granular systems

The similarity in mechanical properties of dense active matter and sheared amorphous solids has been noted in recent years without a rigorous examination of the underlying mechanism. We develop a mean-field model that predicts that their critical behavior-as measured by their avalanche statistics-should be equivalent in infinite dimensions up to a rescaling factor that depends on the correlation length of the applied field. We test these predictions in two dimensions using a numerical protocol, termed "athermal quasistatic random displacement," and find that these mean-field predictions are surprisingly accurate in low dimensions. We identify a general class of perturbations that smoothly interpolates between the uncorrelated localized forces that occur in the high-persistence limit of dense active matter and system-spanning correlated displacements that occur under applied shear. These results suggest a universal framework for predicting flow, deformation, and failure in active and sheared disordered materials.

Numerical solution of the dynamical mean field theory of infinite-dimensional equilibrium liquids

We present a numerical solution of the dynamical mean field theory of infinite-dimensional equilibrium liquids established by Maimbourg et al. [Phys. Rev. Lett. 116, 015902 (2016)]. For soft sphere interactions, we obtain the numerical solution by an iterative algorithm and a straightforward discretization of time. We also discuss the case of hard spheres for which we first derive analytically the dynamical mean field theory as a non-trivial limit of that of soft spheres. We present numerical results for the memory function and the mean square displacement. Our results reproduce and extend kinetic theory in the dilute or short-time limit, while they also describe dynamical arrest toward the glass phase in the dense strongly interacting regime.

Active glasses

Active glassy matter has recently emerged as a novel class of non-equilibrium soft matter, combining energy-driven, active particle movement with dense and disordered glass-like behavior. Here we review the state-of-the-art in this field from an experimental, numerical, and theoretical perspective. We consider both non-living and living active glassy systems, and discuss how several hallmarks of glassy dynamics (dynamical slowdown, fragility, dynamical heterogeneity, violation of the Stokes-Einstein relation, and aging) are manifested in such materials. We start by reviewing the recent experimental evidence in this area of research, followed by an overview of the main numerical simulation studies and physical theories of active glassy matter. We conclude by outlining several open questions and possible directions for future work.

On mean-field theories of dynamics in supercooled liquids

We develop a hybrid numerical approach to extract the exact memory function K(t) of a tagged particle in three-dimensional glass-forming liquids. We compare the behavior of the exact memory function to two mean-field approaches, namely, the standard mode-coupling theory and a recently proposed ansatz for the memory function that forms the basis of a new derivation of the exact form of K(t) for a fluid with short-ranged interactions in infinite dimensions. Each of the mean-field functions qualitatively and quantitatively share traits with the exact K(t), although several important quantitative differences are manifest.