A D Bruce

A study of the multi-canonical Monte Carlo method

We present a study of the multi-canonical Monte Carlo method which constructs and exploits Monte Carlo procedures that sample across an extended space of macrostates. We examine the strategies by which the sampling distribution can be constructed, showing, in particular, that a good approximation to this distribution may be generated efficiently by exploiting measurements of the transition rate between macrostates, in simulations launched from sub-dominant macrostates. We explore the utility of the method in the measurement of absolute free energies, and how it compares with traditional methods based on path integration. We present new results revealing the behaviour of the magnetization distribution of a critical finite-sized magnet, for magnetization values extending from the scaling region all the way to saturation.

Dynamics and statistical mechanics of the Hopfield model

The authors present a study of the Hopfield model of the memory characteristics of a network of interconnected two-state neuron variables. The fraction of nominated configurations which the model stores without error is calculated analytically as a function of the number, N, of neurons and the number, n, of the nominated configurations. The calculation is tested by computer simulation. The noise-free (zero-temperature) phase diagram of the model is determined within a replica-symmetric solution of the mean-field equations. The model exhibits a phase transition at alpha ( identical to n/N)= alpha c approximately=0.069; at this point the thermodynamic states having macroscopic overlap with the nominated configurations disappear, implying a discontinuous change in the fraction of bits (of any nominated configuration) recalled correctly. Large scale Monte Carlo simulations using a distributed array processor provide some support for the existence of a phase transition close to the predicted value.

Universal configurational structure in two-dimensional scalar models

Monte Carlo methods are used to explore the universal configurational structure of two-dimensional spin- 1/2, spin-1 and border- phi 4 models. Comparison of spin- 1/2 and spin-1 data provides evidence that the magnetisation distribution (effectively the Helmholtz free-energy function) and its coupling derivative (effectively the internal-energy function) constitute readily accessible signatures of a universality class. It is shown that, when allowance is made for relatively large corrections-to-scaling effects, the behaviour of the border- phi 4 model may be satisfactorily matched to that of the other two models, substantiating the view that the border model does indeed belong to the Ising universality class.

Crossover behaviour and effective critical exponents in isotropic and anisotropic Heisenberg systems

Renormalisation group methods are used to determine, to second order in epsilon =4-d, the scaling function describing the crossover from Gaussian to Heisenberg behaviour in the susceptibility of an isotropic n-component spin system. The results are used in conjunction with an earlier Feynman graph calculation to obtain an O( epsilon 2) representation of the n-to-m-component susceptibility crossover function, and the corresponding effective exponents, for an anisotropic n-component system.

Droplet theory in low dimensions: Ising systems in zero field

The authors develop a theory of the universal configurational physics underlying critical point phenomena in the Ising universality class. The theory is formally justifiable in d=1+ epsilon dimensions and may be regarded as the natural continuation of the kink-based theory of one dimension, which it incorporates as a limiting case. In d=1+ epsilon the configurational building block is the droplet. The typical droplet is not spherical and the many-droplet assembly is not dilute: the implied problems are handled with renormalisation group methods. It is found that droplet shape fluctuation effects control the correlation length exponent nu , while the nesting of droplets within droplets control the order parameter exponent beta . The exponents nu and beta thus effectively define, respectively, the fractal dimensions of the droplet surface and the droplet volume. The theory is used to determine and illuminate the critical behaviour of further quantities including the free energy, the susceptibility, the droplet number distribution and the distribution of the intra-droplet order.

Learning and memory properties in fully connected networks

This paper summarises recent results of theoretical analysis and numerical simulation, in fully connected networks of the Little‐Hopfield class. The theoretical analysis is based on methods of statistical mechanics as applied to spin‐glass problems, and the numerical work involves massively parallel simulations on the ICL Distributed Array Processor (DAP). Specific applications include: (i) exact results for the fraction of nominal vectors which are perfectly stored by the usual Hebbian rule; (ii) a numerical estimate of the position of the second phase transition in the Hopfield model, at which there is effectively total loss of memory capacity; (iii) a numerical study of the nature of the spurious states in the model; (iv) an exploration of the performance of a learning algorithm, including the exact storage of up to 512 (random) nominal vectors in a 512 node model; (v) a theoretical study of the phase transitions in generalizations where the energy function is a monomial in the state vectors.

Droplet theory in low dimensions: Ising systems in an ordering field

The authors extend the microscopic droplet theory of Ising systems, developed recently, to incorporate the effects of an ordering field. The theory yields a free energy which is the solution of a renormalisation group equation describing droplet nesting, and which has a full scaling form. The behaviour near the coexistence curve is investigated and found to display the essential singularity suggested by primitive droplet models, but with parameters renormalised in a physically intelligible way. The droplet population function implied by the theory is parametrised in a simple way, in space dimension d=2, to yield predictions for various thermodynamic properties, including the equation of state, which are found to be in fair accord with series-based results. The structure of the field-dependent droplet number distributions is investigated and contrasted with the forms assumed in phenomenological droplet theories. The domain of validity of the theory is assessed. It is concluded that the theory fails to describe (sufficiently) large droplets in the presence of a finite field, and is thus in principle trustworthy only in a region close to the coexistence curve.

Droplet theory of correlations in ordered phases

The explicit droplet theory of low-dimensional phase transitions, developed recently, is extended to yield a description of the pair correlation function for the universality class of q-state Potts models in the ordered phase and in zero external field. It is shown that, in d=1+ epsilon dimensions, the short distance behaviour of the correlation function and in particular, the exponent eta , are controlled by nested nearly spherical droplets. In contrast, it is argued that the spatial dependence of the large distance behaviour, in d=2 or above, is controlled by highly anisotropic droplets, thus illuminating the Widom relation, sigma xi d-1 approximately= constant, linking the dimensionless surface tension sigma and the bulk correlation length xi . In the particular case of d=2 the statistical weight of the relevant droplets is determined as that of two appropriately interacting strings, in the spirit of recent independent, arguments by Abraham and by Fisher: a non-Ornstein-Zernike correlation function prefactor, and the result of sigma xi =1/2 follow in accord with exact results for the q=2 (Ising) case, and with implications for the q to 1 (percolation) problem.

Applications of Parallel Computing in Condensed Matter Physics

Computational methods permit the detailed study of microscopic properties and their macroscopic consequences in a host of problems in physics which may be inaccessible to direct experimental study and too complex for theoretical analysis. Reliable calculations from first principles, however, require enormous computing resources. In this talk we describe how parallel computing can provide a practical and cost-effective solution to this problem, illustrating the key ideas with examples from Monte Carlo and molecular dynamics simulations, electronic structure calculations and the analysis of experimental data.

Time coarse graining near critical points

The authors investigate the statistical properties of local variables, coarse grained in time, in systems undergoing continuous phase transitions. A general theory is developed which suggests universal scaling behaviour in the limit of large coarse-graining times. The predictions are substantiated with Monte Carlo studies of two-dimensional scalar models. A simple fractal model of the time profile of the ordering variable is developed which captures and illuminates its timescale-invariant character. The theory is used to study the critical behaviour of local resonance lineshapes. In the two-dimensional case studied explicitly the critical slowing down is shown to drive a crossover to a split slow-motion lineshape.