We study the large deviations of the magnetization at some finite time in the Curie-Weiss Random Field Ising Model with parallel updating. While relaxation dynamics in an infinite time horizon gives rise to unique dynamical trajectories (specified by initial conditions and governed by first-order dynamics of the form m(t+1)=f(m(t)), we observe that the introduction of a finite time horizon and the specification of terminal conditions can generate a host of metastable solutions obeying second-order dynamics. We show that these solutions are governed by a Newtonian-like dynamics in discrete time which permits solutions in terms of both the first order relaxation ("forward") dynamics and the backward dynamics m(t+1)= f^{−1}(m(t)). Our approach allows us to classify trajectories for a given final magnetization as stable or metastable according to the value of the rate function associated with them. We find that in analogy to the Freidlin-Wentzell description of the stochastic dynamics of escape from metastable states, the dominant trajectories may switch between the two types (forward and backward) of first-order dynamics.
We describe the salient ideas of the equilibrium ensemble approach to disordered systems, paying due attention to the appearance of non-Gibbsian measures. A canonical scheme of approximations - constrained annealing - is described and characterised in terms of a Gibbs' variational principle for the free energy functional. It provides a family of increasing exact lower bounds of the quenched free energy of disordered systems, and is shown to avoid the use of non-Gibbsian measures. The connection between the equilibrium ensemble approach and conventional low-concentration expansions or perturbation expansions about ordered reference systems is also explained. Finally applications of the scheme to a number of disordered Ising models and to protein folding are briefly reviewed.
We reply to a comment of Van Enter, Kuelske and Maes Phys. Rev. Lett. 84, 6134 (2000) on our letter "Critical Behavior of the Randomly Spin-Diluted 2-d Ising Model - A Grand Ensemble Approach", Phys. Rev. Lett. 73, 2268-2271 (1994).
The equilibrium ensemble approach to disordered systems is used to investigate the critical behaviour of the two dimensional Ising model in presence of quenched random site dilution. The numerical transfer matrix technique in semi-infinite strips of finite width, together with phenomenological renormalization and conformal invariance, is particularly suited to put the equilibrium ensemble approach to work. A new method to extract with great precision the critical temperature of the model is proposed and applied. A more systematic finite-size scaling analysis than in previous numerical studies has been performed. A parallel investigation, along the lines of the two main scenarios currently under discussion, namely the logarithmic corrections scenario (with critical exponents fixed in the Ising universality class) versus the weak universality scenario (critical exponents varying with the degree of disorder), is carried out. In interpreting our data, maximum care is constantly taken to be open in both directions. A critical discussion shows that, still, an unambiguous discrimination between the two scenarios is not possible on the basis of the available finite size data.
An outline of Morita's equilibrium ensemble approach to disordered systems is given, and hitherto unnoticed relations to other, more conventional approaches in the theory of disordered systems are pointed out. It is demonstrated to constitute a generalization of the idea of grand ensembles and to be intimately related also to conventional low--concentration expansions as well as to perturbation expansions about ordered reference systems. Moreover, we draw attention to the variational content of the equilibrium ensemble formulation. A number of exact results are presented, among them general solutions for site- and bond- diluted systems in one dimension, both for uncorrelated, and for correlated disorder.
The critical behaviour of the 2--d spin-diluted Ising model is investigated by a new method which combines a grand ensemble approach to disordered systems with phenomenological renormalization. We observe a continuous variation of critical exponents with the density $\rho$ of magnetic impurities, respecting, however, weak universality in the sense that $\eta$ and $\gamma/\nu$ do {\it not\/} depend on $\rho$ while $\gamma$ and $\nu$ separately do. Our results are in complete agreement with a recent Monte--Carlo study.