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.
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.