A multineuron interaction model for neural networks

A multineuron interaction model (RS model) with an energy function given by the product of the squared distances in phase space between the state of the net and the stored patterns is studied in detail within a mean-field approach. Two limits are considered: when the patterns and antipatterns are stored (as in the Hopfield model), PAS case, and when only the patterns are taken into account, OPS case. TheT=0 solutions for the proper memories are exactly obtained for all finite values of, as a consequence of the energy function: whenever one of the overlaps is exactly one the corresponding equations decouple and no configuration average is required. Special interest is focused on the OPS situation, which presents a peculiar phase space topology. On the other hand, the PAS configuration recovers the Hopfield model in the appropriate limit, while keeping associative memory abilities far beyond the critical values of other models when the full Hamiltonian is considered.     

Chaotic dynamics of high-order neural networks

The dynamics of an extremely diluted neural network with high-order synapses acting as corrections to the Hopfield model is investigated. The learning rules for the high-order connections contain mixing of memories, different from all the previous generalizations of the Hopfield model. The dynamics may display fixed points or periodic and chaotic orbits, depending on the weight of the high-order connections , the noise levelT, and the network load, defined as the ratio between the number of stored patterns and the mean connectivity per neuron, =P/C. As in the related fully connected case, there is an optimal value of the weight that improves the storage capacity of the system (the capacity diverges).     

Two time scales and violation of the fluctuation-dissipation theorem in a finite dimensional model for structural glasses

We study the breakdown of fluctuation-dissipation relations between time-dependent density-density correlations and associated responses following a quench in the chemical potential in the frustrated Ising lattice gas. The corresponding slow dynamics is characterized by two well-separated time scales characterized by a constant value of the fluctuation-dissipation ratio. This result is particularly relevant since activated processes dominate the long-time dynamics of the system.     

Thermodynamic Properties of a Simple Model of Like-Charged Attracting Rods

We study the thermodynamic properties of a simple model for the possible mechanism of attraction between like charged rod-like polyions inside a polyelectrolyte solution. We consider two polyions in parallel planes, with Z charges each, in a solution containing multivalent counterion of valence . The model is solved exactly for Z13 for a general angle between the rods and supposing that n counterions are condensed onto each polyion. The free energy has two minima, one at =0 (parallel rods) and another at =/2 (perpendicular rods). The stability of the parallel and perpendicular configurations is analyzed.     

Microscopic models of mode-coupling theory: The F(12) scenario

We provide extended evidence that mode-coupling theory (MCT) of supercooled liquids for the F(12) schematic model admits a microscopic realization based on facilitated spin models with tunable facilitation. Depending on the facilitation strength, one observes two distinct dynamical glass transition lines-continuous and discontinuous-merging at a dynamical tricritical-like point with critical decay exponents consistently related by MCT predictions. The mechanisms of dynamical arrest can be naturally interpreted in geometrical terms: the discontinuous and continuous transitions correspond to bootstrap and standard percolation processes, in which the incipient spanning cluster of frozen spins forms either a compact or a fractal structure, respectively. Our cooperative dynamical facilitation picture of glassy behavior is complementary to the one based on disordered systems and can account for higher-order singularity scenarios in the absence of a finite temperature thermodynamic glass transition. We briefly comment on the relevance of our results to finite spatial dimensions and to the F(13) schematic model.     

Relaxation dynamics near the sol–gel transition: From cluster approach to mode-coupling theory

A long standing problem in glassy dynamics is the geometrical interpretation of clusters and the role they play in the observed scaling laws. In this context, the mode-coupling theory (MCT) of type-A transition and the sol–gel transition are both characterized by a structural arrest to a disordered state in which the long-time limit of the correlator continuously approaches zero at the transition point. In this paper, we describe a cluster approach to the sol-gel transition and explore its predictions, including universal scaling laws and a new stretched relaxation regime close to criticality. We show that while MCT consistently describes gelation at mean-field level, the percolation approach elucidates the geometrical character underlying MCT scaling laws.     

Kovacs effect in facilitated spin models of strong and fragile glasses

We investigate the Kovacs (or crossover) effect in facilitated f-spin models of glassy dynamics. Although the Kovacs hump shows a behavior qualitatively similar for all cases we have examined (irrespective of the facilitation parameter f and the spatial dimension d), we find that the dependence of the Kovacs peak time on the temperature of the second quench allows to distinguish among different microscopic mechanisms responsible for the glassy relaxation (e.g. cooperative vs defect diffusion). We also analyze the inherent structure dynamics underlying the Kovacs protocol, and find that the class of facilitated spin models with d>1 and f>1 shows features resembling those obtained recently in a realistic model of fragile glass forming liquid.     

Simple model for attraction between like-charged polyions

We present a simple model for the possible mechanism of appearance of attraction between like charged polyions inside a polyelectrolyte solution. The attraction is found to be short ranged, and exists only in the presence of multivalent counterions. It is produced by the correlations in layers of condensed counterions surrounding each polyion and is only weakly temperature dependent. We find the attraction to be maximum at zero temperature and dimish as the temperature is raised. The attraction is only possible if the number of condensed counterions exceeds the threshold, , where is the valence of counterions and Z is the polyion charge. Received 10 March 1999 and Received in final form 20 April 1999     

Monte Carlo simulations of two-dimensional hard core lattice gases

Monte Carlo simulations are used to study lattice gases of particles with extended hard cores on a two-dimensional square lattice. Exclusions of one and up to five nearest neighbors (NN) are considered. These can be mapped onto hard squares of varying side length, lambda (in lattice units), tilted by some angle with respect to the original lattice. In agreement with earlier studies, the 1NN exclusion undergoes a continuous order-disorder transition in the Ising universality class. Surprisingly, we find that the lattice gas with exclusions of up to second nearest neighbors (2NN) also undergoes a continuous phase transition in the Ising universality class, while the Landau-Lifshitz theory predicts that this transition should be in the universality class of the XY model with cubic anisotropy. The lattice gas of 3NN exclusions is found to undergo a discontinuous order-disorder transition, in agreement with the earlier transfer matrix calculations and the Landau-Lifshitz theory. On the other hand, the gas of 4NN exclusions once again exhibits a continuous phase transition in the Ising universality class-contradicting the predictions of the Landau-Lifshitz theory. Finally, the lattice gas of 5NN exclusions is found to undergo a discontinuous phase transition.     

Dynamic facilitation picture of a higher-order glass singularity

We show that facilitated spin mixtures with a tunable facilitation reproduce, on a Bethe lattice, the simplest higher-order singularity scenario predicted by the mode-coupling theory (MCT) of liquid-glass transition. Depending on the facilitation strength, they yield either a discontinuous glass transition or a continuous one, with no underlying thermodynamic singularity. Similar results are obtained for facilitated spin models on a diluted Bethe lattice. The mechanism of dynamical arrest in these systems can be interpreted in terms of bootstrap and standard percolation and corresponds to a crossover from a compact to a fractal structure of the incipient spanning cluster of frozen spins. Theoretical and numerical simulation results are fully consistent with MCT predictions.