1,721,040 research outputs found
A Tentative Replica Theory of Glassy Helium 4
No full textWe develop a quantum replica method for interacting particle systems and use it to estimate the location of the glass transition line in Helium 4. Although we do not fully succeed in taking into account all quantum effects, we make a thorough semiclassical analysis. We confirm previous suggestions that quantum fluctuations promote the formation of the glass and give a quantitative estimate of this effect at high density. Finally, we discuss the difficulties that are met when one tries to extend the calculation to the region of low densities and low temperatures, where quantum effects are strong and the semiclassical expansion breaks down
How to iron out rough landscapes and get optimal performances: Averaged gradient descent and its application to tensor PCA
No full textIn many high-dimensional estimation problems the main task consists in minimizing a cost function, which is often strongly non-convex when scanned in the space of parameters to be estimated. A standard solution to flatten the corresponding rough landscape consists in summing the losses associated to different data points and obtaining a smoother empirical risk. Here we propose a complementary method that works for a single data point. The main idea is that a large amount of the roughness is uncorrelated in different parts of the landscape. One can then substantially reduce the noise by evaluating an empirical average of the gradient obtained as a sum over many random independent positions in the space of parameters to be optimized. We present an algorithm, called averaged gradient descent, based on this idea and we apply it to tensor PCA, which is a very hard estimation problem. We show that averaged gradient descent over-performs physical algorithms such as gradient descent and approximate message passing and matches the best algorithmic thresholds known so far, obtained by tensor unfolding and methods based on sum-of-squares
Complexity of energy barriers in mean-field glassy systems
Full text linkWe analyze the energy barriers that allow escapes from a given local minimum in a complex high-dimensional landscape. 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, or transition states, and are associated to barriers. We find that the transition state of lowest energy, important for the activated dynamics at low temperature, is strictly below the "threshold" level above which saddles proliferate. We characterize how the quenched complexity of transition states, important for the activated processes at finite temperature, depends on the energy of the state, the energy of the initial minimum, and the distance between them. The overall picture gained from this study is expected to hold generically for mean-field models of the glass transition
Dynamical instantons and activated processes in mean-field glass models
Full text linkWe focus on the energy landscape of a simple mean-field model of glasses and analyze activated barrier-crossing by combining the Kac-Rice method for high-dimensional Gaussian landscapes with dynamical field theory. In particular, we consider Langevin dynamics at low temperature in the energy landscape of the pure spherical p-spin model. We select as initial condition for the dynamics one of the many unstable index-1 saddles in the vicinity of a reference local minimum. We show that the associated dynamical mean-field equations admit two solutions: one corresponds to falling back to the original reference minimum, and the other to reaching a new minimum past the barrier. By varying the saddle we scan and characterize the properties of such minima reachable by activated barrier-crossing. Finally, using time-reversal transformations, we construct the two-point function dynamical instanton of the corresponding activated process
Activated aging dynamics and effective trap model description in the random energy model
Full text linkWe study the out-of-equilibrium aging dynamics of the random energy model (REM) ruled by a single spin-flip Metropolis dynamics. We focus on the dynamical evolution taking place on time-scales diverging with the system size. Our aim is to show to what extent the activated dynamics displayed by the REM can be described in terms of an effective trap model. We identify two time regimes: the first one corresponds to the process of escaping from a basin in the energy landscape and to the subsequent exploration of high energy configurations, whereas the second one corresponds to the evolution from a deep basin to the other. By combining numerical simulations with analytical arguments we show why the trap model description does not hold in the former but becomes exact in the second
Dynamical mean-field theory and aging dynamics
No full textDynamical mean-field theory (DMFT) replaces the many-body dynamical problem with one for a single degree of freedom in a thermal bath whose features are determined self-consistently. By focusing on models with soft disordered p-spin interactions, we show how to incorporate the mean-field theory of aging within DMFT. We study cases with only one slow time-scale, corresponding statically to the one-step replica symmetry breaking phase, and cases with an infinite number of slow time-scales, corresponding statically to the full replica symmetry breaking (FRSB) phase. For the former, we show that the effective temperature of the slow degrees of freedom is fixed by requiring critical dynamical behavior on short time-scales, i.e. marginality. For the latter, we find that aging on an infinite number of slow time-scales is governed by a stochastic equation where the clock for dynamical evolution is fixed by the change of the effective temperature, hence obtaining a dynamical derivation of the stochastic equation at the basis of the FRSB phase. Our results extend the realm of the mean-field theory of aging to all situations where DMFT holds
Theory of the superglass phase
No full textA superglass is a phase of matter which is characterized at the same time by superfluidity and a frozen amorphous structure. We introduce a model of interacting bosons in three dimensions that displays this phase unambiguously and that can be analyzed exactly or using controlled approximations. Employing a mapping between quantum Hamiltonians and classical Fokker-Planck operators, we show that the ground-state wave function of the quantum model is proportional to the Boltzmann measure of classical hard spheres. This connection allows us to obtain quantitative results on static and dynamic quantum correlation functions. In particular, by translating known results on the glassy dynamics of Brownian hard spheres we work out the properties of the superglass phase and of the quantum phase transition between the superfluid and the superglass phase
Facilitated spin models on Bethe lattice: Bootstrap percolation, mode-coupling transition and glassy dynamics RID C-2086-2008
No full textWe show that facilitated spin models of cooperative dynamics introduced by Fredrickson and Andersen display on Bethe lattices a glassy behaviour similar to the one predicted by the mode-coupling theory of supercooled liquids and the dynamical theory of mean-field disordered systems. At low temperature such cooperative models show a two-step relaxation and their equilibration time diverges at a finite temperature according to a power law. The geometric nature of the dynamical arrest corresponds to a bootstrap percolation process which leads to a phase space organization similar to the one of mean-field disordered systems. The relaxation dynamics after a subcritical quench exhibits aging and converges asymptotically to the threshold states that appear at the bootstrap percolation transition
Marginally stable equilibria in critical ecosystems
Full text linkIn this work we study the stability of the equilibria reached by ecosystems formed by a large number of species. The model we focus on are Lotka-Volterra equations with symmetric random interactions. Our theoretical analysis, confirmed by our numerical studies, shows that for strong and heterogeneous interactions the system displays multiple equilibria which are all marginally stable. This property allows us to obtain general identities between diversity and single species responses, which generalize and saturate May's stability bound. By connecting the model to systems studied in condensed matter physics, we show that the multiple equilibria regime is analogous to a critical spin-glass phase. This relation suggests new experimental ways to probe marginal stability
Confinement as a tool to probe amorphous order
No full textWe study the effect of confinement on glassy liquids using Random First Order Transition theory as framework. We show that the characteristic length-scale above which confinement effects become negligible is related to the point-to-set length-scale introduced to measure the spatial extent of amorphous order in super-cooled liquids. By confining below this characteristic size, the system becomes a glass. Eventually, for very small sizes, the effect of the boundary is so strong that any collective glassy behavior is wiped out. We clarify similarities and differences between the physical behaviors induced by confinement and by pinning particles outside a spherical cavity (the protocol introduced to measure the point-to-set length). Finally, we discuss possible numerical and experimental tests of our predictions
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