Prof. David Sherrington FRS

Abstract:

We report a study of a series of simple model systems with only non-interacting Hamiltonians, and hence simple equilibrium thermodynamics, but with constrained dynamics of a type initially suggested by foams and idealized covalent glasses. We demonstrate that macroscopic dynamical features characteristic of real and more complex model glasses, such as two-time decays in energy and auto-correlation functions, arise from the dynamics and we explain them qualitatively and quantitatively in terms of annihilation-diffusion concepts and theory. The comparison is with strong glasses. We also consider fluctuation-dissipation relations and demonstrate subtleties of interpretation. We find no FDT breakdown when the correct normalization is chosen.

Abstract:

We study the dynamics of a version of the batch minority game, with random external information and with different types of inhomogeneous decision noise (additive and multiplicative), using generating functional techniques à la De Dominicis. The control parameters in this model are the ratio alpha=p/N of the number p of possible values for the external information over the number N of trading agents, and the statistical properties of the agents' decision noise parameters. The presence of decision noise is found to have the general effect of damping macroscopic oscillations, which explains why in certain parameter regions it can effectively reduce the market volatility, as observed in earlier studies. In the limit N-->infinity we (i) solve the first few time steps of the dynamics (for any alpha), (ii) calculate the location alpha(c) of the phase transition (signaling the onset of anomalous response), and (iii) solve the statics for alpha>alpha(c). We find that alpha(c) is not sensitive to additive decision noise, but we arrive at nontrivial phase diagrams in the case of multiplicative noise. Our theoretical results find excellent confirmation in numerical simulations.

Abstract:

In this paper we study a simple spin model which has a non-interacting Hamiltonian but constrained dynamics. The model, which is a simplification of a purely topological cellular model (Davison L and Sherrington D 2000 J. Phys. A: Math. Gen. 338615, Aste T and Sherrington D 1999 J. Phys. A: Math. Gen. 32 7049), displays glassy behaviour, involves activated processes and exhibits two-step relaxation. This is a consequence of the existence of annihilation-diffusion processes on two distinct timescales, one temperature independent and the other an exponential function of inverse temperature. In fact, there are several such inter-coupled microscopic processes and great richness therein. Two versions of the model are considered, one with a single absorbing ground state and the other with a highly degenerate ground state. These display qualitatively similar but quantitatively distinct macroscopic behaviour and related, but different, microscopic behaviour. © 2001 IOP Publishing Ltd.