David Sherrington

Thermal Model for Adaptive Competition in a Market

New continuous and stochastic extensions of the minority game, devised as a fundamental model for a market of competitive agents, are introduced and studied in the context of statistical physics. The new formulation reproduces the key features of the original model, without the need for some of its special assumptions and, most importantly, it demonstrates the crucial role of stochastic decision making. Furthermore, this formulation provides the exact but novel nonlinear equations for the dynamics of the system.

Simple strong glass forming models: mean-field solution with activation

We introduce simple models, inspired by previous models for froths and covalent glasses, with trivial equilibrium properties but dynamical behaviour characteristic of strong glass forming systems. These models are also a generalization of backgammon or urn models to a non-constant number of particles, where entropic barriers are replaced by energy barriers, allowing for the existence of activated processes. We formulate a mean-field version of the models, which keeps most of the features of the finite dimensional ones, and solve analytically the out-of-equilibrium dynamics in the low temperature regime where activation plays an essential role.

Phase Diagram and Storage Capacity of Sequence-Storing Neural Networks

We solve the dynamics of Hopfield-type neural networks which store sequences of patterns, close to saturation. The asymmetry of the interaction matrix in such models leads to violation of detailed balance, ruling out an equilibrium statistical mechanical analysis. Using generating functional methods we derive exact closed equations for dynamical order parameters, viz. the sequence overlap and correlation and response functions. in the limit of an infinite system size. We calculate the time translation invariant solutions of these equations. describing stationary limit-cycles. which leads to a phase diagram. The effective retarded self-interaction usually appearing in symmetric models is here found to vanish, which causes a significantly enlarged storage capacity of αc ≈ 0.269. compared to αc ≈ 0.139 for Hopfield networks storing static patterns. Our results are tested against extensive computer simulations and excellent agreement is found.

Replica-Symmetry Breaking in the Critical Behaviour of the Random Ferromagnet

We study the critical properties of the weakly disordered p-component random Heisenberg ferromagnet. It is shown that if the specific-heat critical exponent of the pure system is positive, the traditional renormalization group (RG) flows at dimensions D=4- epsilon , which are usually considered as describing the disorder-induced universal critical behaviour, are unstable with respect to replica-symmetry breaking (RSB) potentials such as those found in spin glasses. It is demonstrated that the RG flows involving RSB potentials lead to fixed points which have a structure known as the one-step RSB, and there exists a whole spectrum of such fixed points. It is argued that spontaneous RSB can occur due to the interactions of the fluctuating fields with the local non-perturbative degrees of freedom coming from the multiple local minima solutions of the mean-field equations. However, it is not clear whether or not RSB occurs for infinitesimally weak disorder. Physical consequences of these conclusions are discussed.

BZT: A Soft Pseudospin Glass.

In an attempt to understand the origin of relaxor ferroelectricity, it is shown that interesting behavior of the onset of nonergodicity and of precursor nanodomains, found in first-principles simulations of the relaxor alloy Ba(Zr(1-x)Ti(x))O(3), can easily be understood within a simple mapping to a soft pseudospin glass.

Continuous time dynamics of the thermal minority game

We study the continuous time dynamics of the thermal minority game. We find that the dynamical equations of the model reduce to a set of stochastic differential equations for an interacting disordered system with nontrivial random diffusion. This is the simplest microscopic description which accounts for all the features of the system. Within this framework, we study the phase structure of the model and find that its macroscopic properties strongly depend on the initial conditions.

Glass transition in self-organizing cellular patterns

We have considered the dynamical evolution of cellular patterns controlled by a stochastic Glauber process determined by the deviations of local cell topology from that of a crystalline structure. Above a critical temperature evolution is towards a common equilibrium state from any initial configuration, but beneath this temperature there is a dynamical phase transition, with a start from a quasi-random state leading to non-equilibrium glassy freezing whereas an ordered start rests almost unchanged. A temporal persistence function decays exponentially in the high-temperature equilibrating state but has a characteristic slow non-equilibrium ageing-like behaviour in the low-temperature glassy phase.

Landscape paradigms in physics and biology: introduction and overview

A brief introductory overview in general terms is given of concepts, issues and applications of the paradigm of rugged landscapes in the contexts of physics and biology.

Coupled dynamics of fast spins and slow interactions in neural networks and spin systems

The authors examine an Ising spin system in which both spins and the interactions between them may evolve in time, although on disparate timescales, such that the couplings change adiabatically. In thermal equilibrium they find a novel application of the replica method, but for finite replica number, representing the ratio of the temperatures of the spin and interaction systems. Regimes where the motion of the couplings has non-trivial effects are found in addition to those where solely the stochasticity of these interaction weights in significant, and this issue is closely related to the orders of the transitions between the various phases observed. Simulation results lend support to the analysis.

Physics and complexity

This paper is concerned with complex macroscopic behaviour arising in many-body systems through the combinations of competitive interactions and disorder, even with simple ingredients at the microscopic level. It attempts to indicate and illustrate the richness that has arisen, in conceptual understanding, in methodology and in application, across a large range of scientific disciplines, together with a hint of some of the further opportunities that remain to be tapped. In doing so, it takes the perspective of physics and tries to show, albeit rather briefly, how physics has contributed and been stimulated.