The complex dynamics of an increasing number of systems is attributed to the emergence of a rugged energy landscape with an exponential number of metastable states. To develop this picture into a predictive dynamical theory I discuss how to compute the exponentially small probability of a jump from one metastable state to another. This is expressed as a path integral that can be evaluated by saddle-point methods in mean-field models, leading to a boundary value problem. The resulting dynamical equations are solved numerically by means of a Newton-Krylov algorithm in the paradigmatic spherical $p$-spin glass model that is invoked in diverse contexts from supercooled liquids to machine-learning algorithms. I discuss the solutions in the asymptotic regime of large times and the physical implications on the nature of the ergodicity-restoring processes.
We study numerically the out-of-equilibrium dynamics of interfaces at finite temperatures when driven well below the zero-temperature depinning threshold. We go further than previous analysis by including the most relevant non-equilibrium correction to the elastic Hamiltonian. We find that the relaxation dynamics towards the steady-state shows glassy behavior, aging and violation of the fluctuation-dissipation theorem. The interface roughness exponent alpha approx 0.7 is found to be robust to temperature changes. We also study the instantaneous velocity signal in the low temperature regime and…
The description of activated relaxation of glassy systems in the multidimensional configurational space is a long-standing open problem. We develop a phenomenological description of the out-of-equilibrium dynamics of a model with a rough potential energy landscape and we analyse it both numerically and analytically. The model provides an example of dynamics where typical relaxation channels go over finite potential energy barriers despite the presence of less-energy-demanding escaping paths in configurational space; we expect this phenomenon to be also relevant in the thermally activated regim…
We explore several potential issues that have been raised over the years regarding the "entropic droplet" scenario of activated transport in liquids, due to Wolynes and coworkers, with the aim of clarifying the status of various approximations of the random first order transition theory (RFOT) of the structural glass transition. In doing so, we estimate the mismatch penalty between alternative aperiodic structures, above the glass transition; the penalty is equal to the typical magnitude of free energy fluctuations in the liquid. The resulting expressions for the activation barrier and the coo…
Using an effective potential approach, we present a replica instanton theory for the dynamics of entropic droplets in glassy systems. Replica symmetry breaking in the droplet interface leads to a length scale dependent reduction of the droplet surface tension and changes the character of the dynamical heterogeneity and activated dynamics in glasses.
We generalize the force-level, microscopic, Nonlinear Langevin Equation (NLE) theory and its elastically collective generalization (ECNLE theory) of activated dynamics in bulk spherical particle liquids to address the influence of random particle pinning on structural relaxation. The simplest neutral confinement model is analyzed for hard spheres where there is no change of the equilibrium pair structure upon particle pinning. As the pinned fraction grows, cage scale dynamical constraints are intensified in a manner that increases with density. This results in the mobile particles becoming mor…
We analyze the energy barriers that allow escapes from a given local minimum in a mean-field model of glasses. We perform this study by using the Kac-Rice method and computing the typical number of critical points of the energy function at a given distance from the minimum. We analyze their Hessian in terms of random matrix theory and show that for a certain regime of energies and distances critical points are index-one saddles and are associated to barriers. We find that the lowest barrier, important for activated dynamics at low temperature, is strictly lower than the "threshold" level above…
We study the evolution of the maximum energy $E_\max(t)$ reached between time $0$ and time $t$ in the dynamics of simple models with glassy energy landscapes, in instant quenches from infinite temperature to a target temperature $T$. Through a detailed description of the activated dynamics, we are able to describe the evolution of $E_\max(t)$ from short times, through the aging regime, until after equilibrium is reached, thus providing a detailed description of the long-time dynamics. Finally, we compare our findings with numerical simulations of the $p$-spin glass and show how the maximum ene…
We study numerically the Kawasaki dynamics of the 2d Ising model. At large time we recover the coarsening growth as l(t) ~ t^(1/3). At shorter time however, the system enters a metastable glassy regime that displays an extremely slow growth and non-trivial violations of the fluctuation dissipation theorem similar to those observed in spin glasses: this is one of the simplest system where such violations occur. We also consider Potts models, where a similar behavior is observed, and the modelof Shore and Sethna where the domain growth is also slow, but where violations of the fluctuation dissip…
We study the emergence of a crossover from entropically driven to thermally activated dynamics in different versions of the `entropic' phase space model introduced by Barrat and Mézard, and previously considered in the zero temperature limit. We first focus on the low temperature (T < Tg) aging phase of the original model and show that a short time singularity appears in the correlation function for Tg/2 < T < Tg, leading to dynamical ultrametricity at T=Tg. We then consider the finite size version of this model, showing that the long time dynamics is always thermally activated beyond a size d…
The following properties are in the present literature associated with the behavior of super-cooled glass-forming liquids: faster than exponential growth of the relaxation time, dynamical heterogeneities, growing point-to-set correlation length, crossover from mean field behavior to activated dynamics. In this paper we argue that these properties are also present in a much simpler situation, namely the melting of the bulk of an ordered phase beyond a first order phase transition point. This is a promising path towards a better theoretical, numerical and experimental understanding of the above …
We compare dynamical heterogeneities in equilibrated supercooled liquids and in the nonequilibrium glassy state within the framework of the random first order transition theory. Fluctuating mobility generation and transport in the glass are treated by numerically solving stochastic continuum equations for mobility and fictive temperature fields that arise from an extended mode coupling theory containing activated events. Fluctuating spatiotemporal structures in aging and rejuvenating glasses lead to dynamical heterogeneity in glasses with characteristics distinct from those found in the equili…
We study thermally activated, low temperature equilibrium dynamics of elastic systems pinned by disorder using one loop functional renormalization group (FRG). Through a series of increasingly complete approximations, we investigate how the field theory reveals the glassy nature of the dynamics, in particular divergent barriers and barrier distributions controling the spectrum of relaxation times. A naive single relaxation time approximation for each wavevector is found to be unsatisfactory. A second approximation based on a random friction model, yields a size (L) dependent log-normal distrib…
We use dynamic light scattering and numerical simulations to study the approach to equilibrium and the equilibrium dynamics of systems of colloidal hard spheres over a broad range of density, from dilute systems up to very concentrated suspensions undergoing glassy dynamics. We discuss several experimental issues (sedimentation, thermal control, non-equilibrium aging effects, dynamic heterogeneity) arising when very large relaxation times are measured. When analyzed over more than seven decades in time, we find that the equilibrium relaxation time, tau_alpha, of our system is described by the …
In this letter we study a reference model in theoretical ecology, the disordered Lotka-Volterra model for ecological communities, in the presence of finite demographic noise. Our theoretical analysis, which takes advantage of a mapping to an equilibrium disordered system, proves that for sufficiently heterogeneous interactions and low demographic noise the system displays a multiple equilibria phase, which we fully characterize. In particular, we show that in this phase the number of stable equilibria is exponential in the number of species. Upon further decreasing the demographic noise, we un…
The thermally activated creep motion of an elastic interface weakly driven on a disordered landscape is one of the best examples of glassy universal dynamics. Its understanding has evolved over the last 30 years thanks to a fruitful interplay between elegant scaling arguments, sophisticated analytical calculations, efficient optimization algorithms and creative experiments. In this article, starting from the pioneer arguments, we review the main theoretical and experimental results that lead to the current physical picture of the creep regime. In particular, we discuss recent works unveiling t…
This lecture deals with glassy dynamics and aging in disordered systems. Special emphasis is put on dynamic mean field theory. In the first part I present some of the systems of interest, in particular spin-glasses, supercooled liquids and glasses, drift, creep and pinning of a particle in a random potential, neural networks, graph partitioning as an example of combinatorial optimisation, the K-sat problem and the minority game as a model for the behaviour of agents on markets. The second part deals with the dynamics of the spherical p-spin-glass with long ranged interactions. This model is a …
We numerically study the potential energy landscape of a fragile glassy system and find that the dynamic crossover corresponding to the glass transition is actually the effect of an underlying geometric transition caused by a qualitative change in the topological properties of the landscape. Furthermore, we show that the potential energy barriers connecting local glassy minima increase with decreasing energy of the minima, and we relate this behaviour to the fragility of the system. Finally, we analyze the real space structure of activated processes by studying the distribution of particle dis…
The slow relaxation and aging of glassy systems can be modelled as a Markov process on a simplified rough energy landscape: energy minima where the system tends to get trapped are taken as nodes of a random network, and the dynamics are governed by the transition rates among these. In this work we consider the case of purely activated dynamics, where the transition rates only depend on the depth of the departing trap. The random connectivity and the disorder in the trap depths make it impossible to solve the model analytically, so we base our analysis on the spectrum of eigenvalues $λ$ of the …
Activated dynamics in a glassy system undergoing steady shear deformation is studied by numerical simulations. Our results show that the external driving force has a strong influence on the barrier crossing rate, even though the reaction coordinate is only weakly coupled to the nonequilibrium system. This "driven activation" can be quantified by introducing in the Arrhenius expression an effective temperature, which is close to the one determined from the fluctuation-dissipation relation. This conclusion is supported by analytical results for a simplified model system.
We discuss a resource-competition model, which takes the MacArthur's model as a platform, to unveil interesting connections with glassy features and jamming in high dimension. This model presents two qualitatively different phases: a "shielded" phase, where a collective and self-sustained behavior emerges, and a "vulnerable" phase, where a small perturbation can destabilize the system and contribute to population extinction. We first present our perspective based on a strong similarity with continuous constraint satisfaction problems in their convex regime. Then, we discuss the stability in te…
The recent implementation of a swap Monte Carlo algorithm (SWAP) for polydisperse mixtures fully bypasses computational sluggishness and closes the gap between experimental and simulation timescales in physical dimensions $d=2$ and $3$. Here, we consider suitably optimized systems in $d=2, 3,\dots, 8$, to obtain insights into the performance and underlying physics of SWAP. We show that the speedup obtained decays rapidly with increasing the dimension. SWAP nonetheless delays systematically the onset of the activated dynamics by an amount that remains finite in the limit $d \to \infty$. This sh…
Supercooled liquids undergo complicated structural relaxation processes, which have been a long-standing problem in both experimental and theoretical aspects of condensed matter physics. In particular, past experiments universally observed for many types of molecular liquids that relaxation dynamics separated into two distinct processes at low temperatures. One of the possible interpretations is that this separation originates from the two-scale hierarchical topography of the potential energy landscape; however, it has never been verified. Molecular dynamics simulations are a promising approac…
Building on the elastically collective nonlinear Langevin equation theory developed for hard spheres in the preceding paper I, we propose and implement a quasi-universal theory for the alpha relaxation of thermal liquids based on mapping them to an effective hard sphere fluid via the dimensionless compressibility. The result is a zero adjustable parameter theory that can quantitatively address in a unified manner the alpha relaxation time over 14 or more decades. The theory has no singularities above zero Kelvin, and relaxation in the equilibrium low temperature limit is predicted to be of a r…
Particle dynamics in supercooled liquids are often dominated by string-like motions in which lines of particles perform activated hops cooperatively. The structural features triggering these motions, crucial in understanding glassy dynamics, remain highly controversial. We experimentally study microscopic particle dynamics in colloidal glass formers at high packing fractions. With a small polydispersity leading to glass-crystal coexistence, a void in the form of a vacancy in the crystal can diffuse reversibly into the glass and further induces string-like motions. In the glass, a void takes th…
We present a framework for analyzing the rheology of dense driven granular materials, based on a recent proposal of a stress-based ensemble. In this ensemble fluctuations in a granular system near jamming are controlled by a temperature-like parameter, the angoricity, which is conjugate to the stress of the system. In this paper, we develop a model for slowly driven granular materials based on the stress ensemble and the idea of a landscape in stress space. The idea of an activated process driven by the angoricity has been shown by Behringer et al (2008) to describe the logarithmic strengtheni…
We develop a framework for understanding the difference between strong and fragile behavior in the dynamics of glass-forming liquids from the properties of the potential energy landscape. Our approach is based on a master equation description of the activated jump dynamics among the local minima of the potential energy (the so-called inherent structures) that characterize the potential energy landscape of the system. We study the dynamics of a small atomic cluster using this description as well as molecular dynamics simulations and demonstrate the usefulness of our approach for this system. Ma…
Stochastic Beta Relaxation (SBR) is a model for the dynamics of glass- forming liquids close to the glass transition singularity of the idealized mode- coupling theory (MCT) that has been derived from generic MCT-like theories by applying dynamical field-theory techniques. SBR displays a rich phenomenology common to most super-cooled liquids. In its simplest version it naturally explains two prominent features of the dynamical crossover: the change from a power-law to exponential increase in the structural relaxation time and the violation of the Stokes-Einstein relation between diffusion and …
