Mean-field Systems Research Articles

Mean-field Theory and Synchronization in Random Recurrent Neural Networks

In this paper, we first present a new mathematical approach, based on large deviation techniques, for the study of a large random recurrent neural network with discrete time dynamics. In particular, we state a mean field property and a law of large numbers, in the most general case of random models with sparse connections and several populations. Our results are supported by rigorous proofs. Then, we focus our interest on large size dynamics, in the case of a model with excitatory and inhibitory populations. The study of the mean field system and of the divergence of individual trajectories allows to define different dynamical regimes in the macroscopic parameters space, which include chaos and collective synchronization phenomenons. At last, we look at the behavior of a particular finite-size system submitted to gaussian static inputs. The system adapts its dynamics to the input signal, and spontaneously produces dynamical transitions from asynchronous to synchronous behaviors, which correspond to the crossing of a bifurcation line in the macroscopic parameters space.

The role of thefree-energy landscape in the dynamics of mean-field glassy systems

In this paper we give the results of our recent work on thedynamical TAP (Thouless-Anderson-Palmer) approach to mean-fieldglassy systems. Our aim is to clarify the connection between thefree-energy landscape and the out-of-equilibrium dynamics insolvable models.Frequently, qualitative explanations of glassy behaviour are basedon the `free-energy landscape paradigm'. While its relevance forequilibrium properties may be clear, the relationship between thefree-energy landscape and out-of-equilibrium dynamics is not wellunderstood yet. In this paper we clarify this relationship for theclass of spin-glass models which reproduce phenomenologically somefeatures of structural glasses. The method that we use is ageneralization of the dynamics of the Thouless, Anderson and Palmerapproach to the thermodynamics of mean-field spin glasses. Withinthis framework, we show to what extent the dynamics can berepresented as an evolution in the free-energy landscape. Inparticular, the relationship between the long-time dynamics and thelocal properties of the free-energy landscape is explicitly apparentusing this approach.

Error-correcting codes and image restoration with multiple stages of dynamics

We consider the problems of error-correcting codes and image restoration with multiple stages of dynamics. Information extracted from the former stage can be used selectively to improve the performance of the latter one. Analytic results were derived for the mean-field systems using the cavity method. We find that it has the advantage of being tolerant to uncertainties in hyperparameter estimation, as confirmed by simulations.

On the magnetic ordering implemented by the s–d exchange interaction

The RKKY perturbation scheme for the spin 1 2 s–d exchange model is reexamined in the case of a finite number of electrons per impurity. The distribution of impurities is assumed to be random. The free electron gas representing the unperturbed system in the RKKY approach is replaced by a mean-field system h r asymptotically equivalent to the reduced s–d model H r which includes part of the s–d exchange. Below the transition temperature T c of h r the lowest energy levels of the s–d exchange Hamiltonian H K resulting from second-order perturbation theory prove to be ordered in the same manner as those of h r in the limit of weak s–d exchange, favouring the same definite parallel alignment of any pair of impurity spins. It follows therefore that in the low temperature, weak coupling regime, the s–d model exhibits ferromagnetic ordering of impurities. At temperatures above T c there is no ordering and impurity–impurity exchange has the RKKY form. These observations are consistent with the presence of a low-temperature ferromagnetic phase in numerous compounds with s–d exchange interaction.

Statistical mechanics of image restoration and error-correcting codes.

We develop a statistical-mechanical formulation for image restoration and error-correcting codes. These problems are shown to be equivalent to the Ising spin glass with ferromagnetic bias under random external fields. We prove that the quality of restoration/decoding is maximized at a specific set of parameter values determined by the source and channel properties. For image restoration in a mean-field system a line of optimal performance is shown to exist in the parameter space. These results are illustrated by solving exactly the infinite-range model. The solutions enable us to determine how precisely one should estimate unknown parameters. Monte Carlo simulations are carried out to see how far the conclusions from the infinite-range model are applicable to the more realistic two-dimensional case in image restoration.

Small random perturbation of a classical mean field model

The thermodynamic limit of a classical mean field system perturbed by a small Sherrington–Kirkpatrick term is derived. It is proved to exhibit behaviors that may strongly differ from the usual ones. Examples are given showing that every situation is likely to occur. The main tools that are used are large deviation techniques and conditioning.

Quenched complexity of the mean-fieldp-spin spherical model with external magnetic field

We consider the p-spin spherical spin-glass model in the presence of an external magnetic field as a general example of a mean-field system where a one-step replica symmetry breaking (1-RSB) occurs. In this context we compute the complexity of the Thouless-Anderson-Palmer states, performing a quenched computation. We find what the general connection is between this method and the standard static 1-RSB one, formulating a clear mapping between the parameters used in the two different calculations. A dynamical analysis of the model confirms the validity of our results.

Traveling density wave models for earthquakes and driven threshold systems

Driven threshold systems are now used to model sandpiles, earthquakes, magnetic depinning transitions, integrate-and-fire neural networks, and driven foams. We analyze a physically motivated model which has many of the same properties as the hard threshold models, but in which all of the nonequilibrium physics is obtained from a Lyapunov functional. The ideas apply to mean-field systems, and lead to a number of predictions, including scaling exponents and metastable lifetimes for nucleating droplets. The former predictions are supported, for example, by data observed for earthquake fault systems. An interesting consequence of the model is that time appears as a scaling field, leading to temporal scaling laws similar to those observed in nature.

Quantum critical effects in mean-field glassy systems

We consider the effects of quantum fluctuations in mean-field quantum spin-glass models with pairwise interactions. We examine the nature of the quantum glass transition at zero temperature in a transverse field. In models (such as the random orthogonal model) where the classical phase transition is discontinuous an analysis using the static approximation reveals that the transition becomes continuous at zero temperature.

Multilevel bilinear systems of stochastic differential equations

A multilevel bilinear system of stochastic differential equations is a multilevel mean field system in which the drift term is also linear. Two kinds of parameters coexist in this model: the rate of spatial mixing and the noise intensity. The parameter space is partitioned into three regions that correspond to qualitatively different system behaviours also known as subcritical, critical and supercritical states. We obtain a complete description of the subcritical state and, particularly, the limiting behavior of the process when we rescale the time. We develop a new technique involving fractional moments which allows us to describe partially the supercritical state. The critical state is a very difficult one and although there some open questions remain, we have obtained rigorous partial results.

Liapunov functionals and large-time-asymptotics of mean-field nonlinear Fokker-Planck equations

Abstract Large-time asymptotics of the solutions of time-dependent mean-field nonlinear Fokker-Planck systems are analyzed by exploiting the properties of Liapunov functionals (relative entropies). We first discuss the scalar Desai-Zwanzig model and then generalize the techniques to a general class of mean-field oscillator equations. Systems of this type are relevant in the modeling of synchronization of a large population of coupled nonlinear oscillators in the presence of external or thermal white noise.

Global cocycle dynamics for infinite mean field quantum systems interacting with the boson gas

In the framework of operator algebraic quantum statistical mechanics, the global nonequilibrium dynamics for a general class of interactions of infinite mean field quantum lattice systems with the boson field is investigated. The associated interaction operators consist of arbitrary powers of the collective density operators of the mean field system and are linear with respect to the bosonic field operators. Instead of the usual perturbation expansions, here the interacting dynamics are studied by means of cocycle techniques. The cocycle methods are more appropriate to the considered class of interactions, and lead to explicit closed expressions for the dynamical automorphism groups. The cocycle equations connect the classical, collective dynamical behavior of the mean field system with the one-boson dynamics. In physical applications such systems are due to collectively ordered finite-level atoms or the Josephson junction in the thermodynamic limit weakly interacting with the electromagnetic field.

Clustering and relaxation in Hamiltonian long-range dynamics.

We study the dynamics of a fully coupled network of N classical rotators, which can also be viewed as a mean-field XY Heisenberg (HMF) model, in the attractive (ferromagnetic) and repulsive (antiferromagnetic) cases. The exact free energy and the spectral properties of a Vlasov-Poisson equation give hints on the values of dynamical observables and on time relaxation properties. At high energy (high temperature T) the system relaxes to Maxwellian equilibrium with vanishing magnetization, but the relaxation time to the equilibrium momentum distribution diverges with N as ${\mathit{NT}}^{2}$ in the ferromagnetic case and as ${\mathit{NT}}^{3/2}$ in the antiferromagnetic case. The N dependence of the relaxation time is suggested by an analogy of the HMF model with gravitational and charged sheets dynamics in one dimension and is verified in numerical simulations. Below the critical temperature the ferromagnetic HMF mode shows a collective phenomenon where the rotators form a drifting cluster; we argue that the drifting speed vanishes as ${\mathit{N}}^{\mathrm{\ensuremath{-}}1/2}$ but increases as one approaches the critical point (a manifestation of critical slowing down). For the antiferromagnetic HMF model a two-cluster drifting state with zero magnetization forms spontaneously at very small temperatures; at larger temperatures an initial density modulation produces this state, which relaxes very slowly. This suggests the possibility of exciting magnetized states in a mean-field antiferromagnetic system.

Static and kinetic aspects of the ordering transition in thin films of diblock copolymers.

Using time-resolved X-ray reflection experiments, we have investigated the surface structure of thin films of a symmetric diblock copolymer of polystyrene-b-polyisoprene (M w = 15 700). The film thickness ranges from 20 to 80 nm. The ordering transition is observed in a temperature-driven experiment. We find the transition temperature to be raised above its bulk value. It is close to the transition temperature of the ideal mean-field system. The evolution of order proceeds through a roughening of the film surface and subsequently the formation of an incomplete top layer. In atomic force microscopy measurements this layer is shown to consist of islands and depressions. The thickness of this layer is precisely one bulk lamellar period L = 16.2 nm. The lamellae are oriented parallel to the substrate. The base layer is expanded in thickness to 20 nm. The formation of the ordered structure follows the same time constants that were found for the long-range ordering process in the bulk of this polymer.

Invasion, stability and evolution to criticality in spatially extended, artificial host—pathogen ecologies

We consider an individual-based spatial model of a generic host-pathogen system and explore the differences between such models and mean-field systems. We find a range of new dynamical and evolutionary phenomena, in particular: (i) in this system, selective pressure is substantially reduced compared with the corresponding mean-field models, and artificial suppression of the pathogen population leads to faster evolution and reduces evolutionary stability; (ii) unlike the mean-field models, there exists a critical transmissibility $\tau \_{\text{c}}$ above which the pathogen dies out; and (iii) the system displays self-evolved criticality. If the transmissibility $\tau $ is allowed to mutate, it evolves to the critical value $\tau \_{\text{c}}$. Thus the system evolves to put itself at the boundary at which it can exist. Observations of the individual-based spatial model motivate an explanation for these phenomena in terms of the dynamics of host patches involving their connections and disconnections. We therefore construct a patch model of this and show that this simplified model behaves in a similar way to the individual-based spatial model.

Self-averaging in a class of generalized Hopfield models

We prove the almost sure convergence to zero of the fluctuations of the free energy in a class of disordered mean-field spin systems that generalize the Hopfield model in two ways: (i) Multi-spin interactions are permitted and (ii) the random variables xi imu describing the 'patterns' can have arbitrary distributions with mean zero and finite (4+ epsilon )th moments. The number of patterns, M, is allowed to be an arbitrary multiple of the system size. This generalizes a previous result of Bovier, Gayrard and Picco for the standard Hopfield model, and improves a result of Feng and Tirozzi that required M to be a finite constant. Note that the convergence of the mean of the free energy is not proven.

Constrained Potts mean field systems and their electronic implementation.

The Potts mean field approach for solving combinatorial optimization problems subject to winner-takes-all constraints is extended for problems subject to additional constraints. Extra variables corresponding to the Lagrange multipliers are incorporated into the Potts formulation for the additional constraints to be satisfied. The extended Potts equations are solved by using constrained gradient descent differential systems. This gradient system is proven theoretically to always produce a legal local optimum solution of the constrained combinatorial optimization problems. An analog electronic circuit implementing the present method is designed on the basis of the previous Potts electronic circuit. The performance of the present method is theoretically evaluated for the constrained maximum cut problems. The lower bound of the cut size obtained with the present method is proven to be the same as that of the basic Potts scheme for the unconstrained maximum cut problems.

Do classical spin systems with the same metastable states have identical Hamiltonians?

The question whether systems having the same set of metastable states have identical Hamiltonian and hence identical dynamics is addressed and examined for various classical spin systems. The answer depends on the number of metastable states and on the distribution of the local fields for small fields. The rule that static properties determine dynamical properties is found to be applicable for a large class of random systems, almost any random system in the mean-field limit and many random systems in finite number of dimensions.

Phase separation in deuterated polycarbonate/poly(methylmethacrylate) blend near glass transition temperature

Early stage of spinodal decomposition kinetics of deuterated polycarbonate/poly(methylmethacrylate) blend has been studied by the time resolved small angle neutron scattering measurements. A trapped miscible state was obtained through fast solution casting of film specimens. Time dependence of the structure factor, S(q,t), was measured after the specimen temperature was quickly increased to a final temperature which is above its glass transition temperature, Tg. Thus, the early stage of spinodal decomposition kinetics has been observed starting from the dimension (q−1) comparable to the single chain radius of gyration, Rg, for a binary polymer mixture. The results provide an unequivocal quantitative measure of the virtual structure factor, S(q,∞); the relationship of qm and qc through rate of growth, Cahn-plot analysis, and singularity in S(q,∞); the growth of fluctuation of qRg<1 and intrachain relaxation of qRg≳1; and also a clear proof of the Cahn–Hillard–Cook theory in the early stage of spinodal decomposition of a mean-field system.

A cooperative self‐consistent microscopic theory of thermally induced melting of a repeat sequence DNA polymer

AbstractWe attempt to extend the modified self‐consistent phonon theory to describe thermal fluctuational base‐pair opening of repeat sequence DNA polymers in the helix–coil transition region as well as in the premelting region. A microscopic base‐pair open state is introduced and the effect of this open state is taken into account self‐consistently in a mean field system that models the DNA polymer. Our analysis indicates the structure of this open state changes with temperature in such a manner that on average a base pair opens and unstacks with its neighbors more completely as temperature increases. We apply this theory to a homopolymer—poly(dG) · poly(dC) to evaluate the base‐pair opening probability in a temperature range from 273 to 366.5 K. At 366.5 K the system undergoes cooperative melting. Our calculated base‐pair opening probabilities are in general agreement with several experimental estimates at room temperature. The calculated probabilities show typical melting curve behavior at temperatures close to the observed melting temperature. The cooperative modified self‐consistent phonon approximation approach becomes a viable microscopic theory of melting. © 1993 John Wiley & Sons, Inc.