G. Toulouse

Networks of Formal Neurons and Memory Palimpsests

One characteristic behaviour of the Hopfield model of neural networks, namely the catastrophic deterioration of the memory due to overloading, is interpreted in simple physical terms. A general formulation allows for an exploration of some basic issues in learning theory. Two learning schemes are constructed, which avoid the overloading deterioration and keep learning and forgetting, with a stationary capacity.

Theory of the frustration effect. II. Ising spins on a square lattice

For pt.I see Commun. Phys., vol.2, p.115 (1977). Presents numerical results for the ground-state properties of the simplest system showing the frustration effect, Ising spins on a square lattice with a variable concentration x of antiferromagnetic interactions. Ferromagnetism disappears at a concentration xc approximately 0.09. The ground-state energy and degeneracy have markedly different behaviours below and above xc. The authors compare the results with other calculations and stress the advantages of the frustration approach.

Dynamical corrections to the image potential

Abstract The effective potential and force acting on a moving charge near a metal surface are calculated, taking into account dynamical effects due to the finite mass and velocity of the charge.

Perspectives from the theory of phase transitions

SummaryAn illustrated discussion is given of perspectives offered by the general theory of critical phenomena for the many-electron problem.RiassuntoSi fa una discussione illustrativa delle prospettive offerte dalla teoria generale dei fenomeni critici per il problema di molti elettroni.РезюмеВ последнее время достигнут большой успех в теории критичесих явлений и фазвых переходов. В этой работе рассматриваются перспективы, которые открывает общая теория для многоэлектронных проблем.

Nonadiabatic electronic response in atom scattering from metal surfaces

The response of metal electrons to a localized perturbation varying in time is singular; this dynamic property may influence the observability of diffraction peaks in atom scattering from metal surfaces.

Phase transitions for one and zero dimensional systems with short-range forces

It is shown that analytic continuation on n, the number of components of the order parameter, yields phase transitions at finite temperatures for one dimensional systems with short range forces, as soon as n < 1 (for zero dimensional systems, as soon as n < 0). The critical exponents are found to satisfy the scaling laws, with ν = η − 1 for d = 1 and −2 <n < 1. The value n = −2 corresponds to a triple point with different exponents, ν = 12, η = 0. The equations of state are also derived. Two models are solved, yielding comparable results: Stanleys generalized Heisenberg model (which has a transition for 0 <n < 1), and the Ginzburg-Landau-Wilson model. The latter amounts to the study of the n-dimensional isotropic anharmonic oscillator whose properties are shown to depend only on the combination 12n + l, where l is the angular momentum. Some exact eigenvalues, exhibiting universality with respect to anharmonicity, and the associated eigenfunctions, are derived for 12n + l = 0, −1,…. The study of the various types of level crossing and of their existence domains leads to a detailed description of the corresponding phase diagram. Similar properties are also derived for zero dimensional phase transitions.

Theory of the ionization probability for an atom crossing a metal-vacuum surface

Abstract A theory of the ionization probability and its application to dilute alloy systems are presented.

On the connection between spin glasses and gauge field theories

Abstract A simple connection between Ising spin glasses and the Z2 lattice gauge theory, at negative plaquette temperatures, is presented. It is first shown that annealed models give useful lower bounds on the free energy and ground-state energy of spin glasses. However, they have unphysical low temperature properties (e.g. a negative entropy), which are related to a temperature dependence of the frustration. A restricted annealing scheme is presented which remedies this deficiency through the introduction of a pure gauge coupling counterterm. The possible phase diagrams of the lattice gauge system and their relevance to spin glass transitions are discussed.

Information capacity of a perceptron

The authors study the information storage capacity of a simple perceptron in the error regime. For random unbiased patterns the geometrical analysis gives a logarithmic dependence for the information content in the asymptotic limit. In this case, the statistical physics approach, when used at the simplest level of replica theory, does not give satisfactory results. However for perceptrons with finite stability, the information content can be simply calculated with statistical physics methods in a region above the critical storage level, for biased as well as for unbiased patterns.

Topological solitons and graded Lie algebras

This paper is an extension of our previous investigation on the crossing of defects in ordered media. We provide a general mathematical framework, in which the obstructions for crossing p‐dimensional defects with q‐dimensional defects in a (p+q+1) ‐dimensional sample are the brackets of a certain graded Lie algebra (connected with homotopy theory). This ’’confinement mechanism’’ brings together the mathematical structures of two not yet related themes in present‐day physics: solitons and supersymmetry.