Preface page ix 1 Getting Started 1.1 Artificial neural networks 1.2 A simple example 1.3 General setup 1.4 Problems 2 Perceptron Learning-Basics 2.1 Gibbs learning 2.2 The annealed approximation 2.3 The Gardner analysis 2.4 Summary 2.5 Problems 3 A Choice of Learning Rules 3.1 The Hebb rule 3.2 The perceptron rule 3.3 The pseudo-inverse rule 3.4 The adaline rule 3.5 Maximal stability 3.6 The Bayes rule 3.7 Summary 3.8 Problems 4 Augmented Statistical Mechanics Formulation 4.1 Maximal stabilities 4.2 Gibbs learning at non-zero temperature 4.3 General statistical mechanics formulation 4.4 Learning rules revisited 4.5 The optimal potential 4.

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We consider a diluted and nonsymmetric version of the Little-Hopfield model which can be solved exactly. We obtain the analytic expression of the evolution of one configuration having a finite overlap on one stored pattern. We show that even when the system remembers, two different configurations which remain close to the same pattern never become identical. Lastly, we show that when two stored patterns are correlated, there exists a regime for which the system remembers these patterns without being able to distinguish them.

Molecular dynamics simulations are used to generate an ensemble of saddles of the potential energy of a Lennard-Jones liquid. Classifying all extrema by their potential energy u and number of unstable directions k, a well-defined relation k(u) is revealed. The degree of instability of typical stationary points vanishes at a threshold potential energy u(th), which lies above the energy of the lowest glassy minima of the system. The energies of the inherent states, as obtained by the Stillinger-Weber method, approach u(th) at a temperature close to the mode-coupling transition temperature T(c).

As Charles Goodyear discovered in 1839, when he first vulcanised rubber, a macromolecular liquid is transformed into a solid when a sufficient density of permanent crosslinks is introduced at random. At this continuous equilibrium phase transition, the liquid state, in which all macromolecules are delocalised, is transformed into a solid state, in which a nonzero fraction of macromolecules have spontaneously become localised. This solid state is a most unusual one: localisation occurs about mean positions that are distributed homogeneously and randomly, and to an extent that varies randomly from monomer to monomer. Thus, the solid state emerging at the vulcanisation transition is an equilibrium amorphous solid state: it is properly viewed as a solid state that bears the same relationship to the liquid and crystalline states as the spin glass state of certain magnetic systems bears to the paramagnetic and ferromagnetic states, in the sense that, like the spin glass state, it is diagnosed by a subtle order parameter.In this article we give a detailed exposition of a theoretical approach to the physical properties of systems of randomly, permanently crosslinked macromolecules. Our primary focus is on the equilibrium properties of such systems, especially in the regime of Goodyear's vulcanisation transition. This approach rests firmly on techniques from the statistical mechanics of disordered systems pioneered by Edwards and co-workers in the context of macromolecular systems, and by Edwards and Anderson in the context of magnetic systems. We begin with a review of the semi-microscopic formulation of the statistical mechanics of randomly crosslinked macromolecular systems due to Edwards and co-workers, in particular discussing the role of crosslinks as quenched random variables. Then we turn to the issue of order parameters, and review a version capable, inter alia, of diagnosing the amorphous solid state. To develop some intuition, we examine the order parameter in an idealised situation, which subsequently turns out to be surprisingly relevant. Thus, we are motivated to hypothesise an explicit form for the order parameter in the amorphous solid state that is parametrised in terms of two physical quantities: the fraction of localised monomers, and the statistical distribution of localisation lengths of localised monomers. Next, we review the symmetry properties of the system itself, the liquid state and the amorphous solid state, and discuss connections with scattering experiments. Then, we review a representation of the statistical mechanics of randomly crosslinked macromolecular systems from which the quenched disorder has been eliminated via an application of the replica technique. We transform the statistical mechanics into a field-theoretic representation, which exhibits a close connection with the order parameter, and analyse this representation at the saddle-point level. This analysis reveals that sufficient crosslinking causes an instability of the liquid state, this state giving way to the amorphous solid s...