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Fluids with quenched disorder: Scaling of the free energy barrier near critical points

In the context of Monte Carlo simulations, the analysis of the probability distribution $P_L(m)$ of the order parameter $m$, as obtained in simulation boxes of finite linear extension $L$, allows for an easy estimation of the location of the critical point and the critical exponents. For Ising-like systems without quenched disorder, $P_L(m)$ becomes scale invariant at the critical point, where it assumes a characteristic bimodal shape featuring two overlapping peaks. In particular, the ratio between the value of $P_L(m)$ at the peaks ($P_{L, max}$) and the value at the minimum in-between ($P_{L, min}$) becomes $L$-independent at criticality. However, for Ising-like systems with quenched random fields, we argue that instead $ΔF_L := \ln (P_{L, max} / P_{L, min}) \propto L^θ$ should be observed, where $θ>0$ is the "violation of hyperscaling" exponent. Since $θ$ is substantially non-zero, the scaling of $ΔF_L$ with system size should be easily detectable in simulations. For two fluid models with quenched disorder, $ΔF_L$ versus $L$ was measured, and the expected scaling was confirmed. This provides further evidence that fluids with quenched disorder belong to the universality class of the random-field Ising model.