Monte Carlo methods, Irreversible Markov chains, Spin glasses, Dynamical mean field theory, Mode coupling theory, Non-Reciprocal interactions, Glass transition, Numerical simulations, Stochastic processes

In silico sampling of a target distribution for a system of many interacting degrees of freedom is a ubiquitous task in natural sciences. The relaxation time, the typical time scale over which convergence is achieved, becomes dauntingly large in some relevant problems, ranging from protein folding to neural networks training . The hardest instance of this problem in condensed matter is presented by disordered systems, from spin glasses to supercooled liquids, whose relaxation time grows by many orders of magnitude upon a mild change in temperature or density. The way-to-go paradigm to reduce the relaxation time relies on designing alternative, nonphysical dynamical evolution rules that sample the target distribution, which is achieved by preserving time reversibility. Another route, which has its birthplace in applied mathematics, consists in supplementing with a nonequilibrium drive an otherwise equilibrium stochastic process, which can be rewarded with a reduction of the relaxation time. This thesis addresses the quantitative performance of irreversible dynamics that sample the Boltzmann distribution in different models and system displaying glassy dynamical slowdown. In Chapter 1 we describe different ways by which the dynamics can be driven out of equilibrium while preserving the target distribution, and we identify a minimal implementation of the nonequilibrium drive, achieved by supplementing an equilibrium overdamped Langevin dynamics with forces transverse to the energy gradient. In Chapter 2 we move to the dynamics of transverse forces for a single particle evolving in an external potential. In Chapter 3, we study the dynamics of a mean field spin glass with transverse forces. We quantify and physically understand the speedup by means of novel cross correlation function and of a modified fluctuation dissipation theorem. In Chapter 4 we bring transverse forces to work in dense liquids through numerical simulations in finite dimensions. We discover that the speedup is a nonmonotonus function of the temperature and that, as we enter the region of glassy dynamics, the efficiency of transverse forces decreases. Microscopically, we find that in this region the dynamical pathways unlocked by transverse forces fold themselves in circular orbits. We characterize these trajectories by measuring the odd diffusion constant, a transport coefficient now permitted by transverse forces. In Chapter 5 and 6 we rationalize our numerical investigation, developing a dynamical mean field theory and a mode coupling theory specifically tailored for transverse forces. We address the emergence of several transport coefficients (diffusivity, viscosity, mobility), with a focus on their odd components, and determine the asymptotic behavior of the speedup as transverse forces are made increasingly strong. In Chapter 7, we probe the performance of the Event-Chain Monte Carlo algorithm, a nonequilibrium sampling method that performs driven, collective translations of chains of particles, in a polydisperse glass former of hard disks. At all densities explored, ECMC maintains an edge over the equilibrium Metropolis algorithm. Echoing what we found for transverse forces, the efficiency of ECMC decreases as the density of the system increases. We then propose a novel algorithm, collective Swap (cSwap), which performs out of equilibrium, collective swaps of particle diameters. We show that it outperforms state-of-the art Mote Carlo algorithms, and that its efficiency increases with the density within our observation window. By combining cSwap and ECMC, we achieve a speedup of 40 over state of the art methods. As an application, we use the novel algorithm to produce stable jammed packings at remarkably high densities. In the conclusions, we summarize the contributions developed in the previous chapter, and propose future research directions.