Barriers between metastable states in the p-spin spherical model (original) (raw)
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Journal of Physics A: Mathematical and General, 1997
In this paper we introduce a three replica potential useful to examine the structure of metastable states above the static transition temperature T c , in the spherical p-spin model. Studying the minima of the potential we are able to find which is the distance between the nearest equilibrium and local equilibrium states, obtaining in this way information on the dynamics of the system. Furthermore, the analysis of the potential at the dynamical transition temperature T d suggests that equilibrium states are not randomly distributed in the phase space.
On the Number of Metastable States in Spin Glasses
Europhysics Letters (EPL), 1995
In this letter, we show that the formulae of Bray and Moore for the average logarithm of the number of metastable states in spin glasses can be obtained by calculating the partition function with m coupled replicas with the symmetry among these explicitly broken according to a generalization of the 'two-group' ansatz. This equivalence allows us to find solutions of the bm equations where the lower 'band-edge' free energy equals the standard static free energy. We present these results for the Sherrington-Kirkpatrick model, but we expect them to apply to all mean-field spin glasses.
2012
In these notes, we continue our investigation of classical toy models of disordered statistical mechanics, through techniques recently developed and tested mainly on the paradigmatic Sherrington-Kirkpatrick spin glass. Here, we consider the p-spin-glass model with Ising spins and interactions drawn from a normal distribution N [0, 1]. After a general presentation of its properties (e.g. self-averaging of the free energy, existence of a suitable thermodynamic limit), we study its equilibrium behavior within the Hamilton-Jacobi framework and the smooth cavity approach. Through the former we find both the RS and the 1-RSB expressions for the free-energy, coupled with their self-consistent relations for the overlaps. Through the latter, we recover these results as irreducible expression, and we study the generalization of the overlap polynomial identities suitable for this model; a discussion on their deep connection with the structure of the internal energy and the entropy closes the investigation.
Journal of Statistical Physics, 2006
The aim of this paper is to discuss the main ideas of the Talagrand proof of the Parisi Ansatz for the free-energy of Mean Field Spin Glasses with a physicist's approach. We consider the case of the spherical p-spin model, which has the following advantages: 1) the Parisi Ansatz takes the simple "one step replica symmetry breaking form", 2) the replica free-energy as a function of the order parameters is simple enough to allow for numerical maximization with arbitrary precision. We present the essential ideas of the proof, we stress its connections with the theory of effective potentials for glassy systems, and we reduce the technically more difficult part of the Talagrand's analysis to an explicit evaluation of the solution of a variational problem.
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We discuss the metastate, a probability measure on thermodynamic states, and its usefulness in addressing difficult questions pertaining to the statistical mechanics of systems with quenched disorder, in particular short-range spin glasses. The possible low-temperature structures of realistic (i.e., short-range) spin glass models are described, and a number of fundamental open questions are presented.
Some comments on the Sherrington-Kirkpatrick model of spin glasses
Communications in Mathematical Physics, 1987
In this paper the high-temperature phase of general mean-field spin glass models, including the Sherrington-Kirkpatrick (SK) model, is analyzed. The free energy in zero magnetic field is calculated explicitly for the SK model, and uniform bounds on quenched susceptibilities are established. It is also shown that, at high temperatures, mean-field spin glasses are limits of shortrange spin glasses, as the range of the interactions tends to infinity.
Energy barriers in SK spin-glass model
Journal de Physique, 1989
2014 Nous étudions les hauteurs de barrière séparant des états métastables pour le modèle de verre de spin avec symétrie d'Ising et portée infinie. Une configuration de barrière correspond à un col de {mi, i = 1, ..., N, -1 ~ mi ~ 1} de la surface d'énergie qui interpole de façon régulière l'énergie dans l'hypercube. Pour des barrières d'énergie faibles, qui sont importantes pour la dynamique, le nombre de directions descendantes au col est fini dans la limite N ~ ~. Nous trouvons que ces directions sont contenues dans un sous-espace linéaire de l'hypercube [20141,1]N engendré par les directions pour lesquelles 03A3j Jij mj = 0.
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In talk I will review the theoretical results that have been obtained for spin glasses, paying a particular attention to finite dimensional spin glasses. I will concentrate my attention on the formulation of the mean field approach and on its numerical and experimental verifications. I will mainly considered equilibrium properties at zero magnetic field, where the situation is clear and it should be not controversial. I will present the various hypothesis at the basis of the theory and I will discuss their physical status.
An investigation of the hidden structure of states in a mean field spin glass model
1997
We study the geometrical structure of the states in the low temperature phase of a mean field model for generalized spin glasses, the p-spin spherical model. This structure cannot be revealed by the standard methods, mainly due to the presence of an exponentially high number of states, each one having a vanishing weight in the thermodynamic limit. Performing a purely entropic computation, based on the TAP equations for this model, we define a constrained complexity which gives the overlap distribution of the states. We find that this distribution is continuous, non-random and highly dependent on the energy range of the considered states. Furthermore, we show which is the geometrical shape of the threshold landscape, giving some insight into the role played by threshold states in the dynamical behaviour of the system.
Distribution of metastable states of spin glasses
Journal of Physics: Conference Series
The complex behavior of systems like spin glasses, proteins or neural networks is typically explained in terms of a rugged energy or fitness landscape. Within the highdimensional conformation space of these systems one finds features like barriers, saddle points, and metastable states whose number-at least in the case of spin glasses-grows exponentially with the size of the system. We propose a novel Monte Carlo sampling algorithm that employs an ensemble of short Markovian chains in order to visit all metastable states with equal probability. We apply this algorithm in order to measure the number of metastable states for the twodimensional and the three-dimensional Edwards-Anderson model and compare with theoretical predictions.