We consider the isotropic-to-nematic transition in liquid crystals confined to aerogel hosts, and assume that the aerogel acts as a random field. We generally find that self-averaging is violated. For a bulk transition that is weakly first-order, the
violation of self-averaging is so severe, even the correlation length becomes non-self-averaging: no phase transition remains in this case. For a bulk transition that is more strongly first-order, the violation of self-averaging is milder, and a phase transition is observed.
As a generic model for liquid-vapour type transitions in random porous media, the Asakura-Oosawa model for colloid-polymer mixtures is studied in a matrix of quenched spheres using extensive Monte Carlo (MC) simulations. Since such systems at critica
lity, as well as in the two-phase region, exhibit lack of self-averaging, the analysis of MC data via finite size scaling requires special care. After presenting the necessary theoretical background and the resulting subtleties of finite size scaling in random-field Ising-type systems, we present data on the order parameter distribution (and its moments) as a function of colloid and polymer fugacities for a broad range of system sizes, and for many (thousands) realizations of the porous medium. Special attention is paid to the connected and disconnected susceptibilities, and their respective critical behavior. We show that both susceptibilities diverge at the critical point, and we demonstrate that this is compatible with the predicted scenario of random-field Ising universality.
