Ordering and demixing transitions in multicomponent Widom-Rowlinson models

Abstract

We use Monte Carlo techniques and analytical methods to study the phase diagram of multicomponent Widom-Rowlinson models on a square lattice:there are M species all with the same fugacity z and a nearest neighbor hard core exclusion between unlike particles. Simulations show that for M between two and six there is a direct transition from the gas phase at z < z d (M ) to a demixed phase consisting mostly of one species at z > z d (M ) while for M ≥ 7 there is an intermediate "crystal phase" for z lying between z c (M ) and z d (M ). In this phase, which is driven by entropy, particles, independent of species, preferentially occupy one of the sublattices, i.e. spatial symmetry

Phase transitions in the multicomponent Widom-Rowlinson model and in hard cubes on the bcc lattice

“…We also find, as in [9], that for large M, z d (M) for the crystal-demixed transition can be computed via Pirogov-Sinai theory [14] yielding,…”

“…This method for locating the transition fugacities was used in our previous studies [8,9] on the square lattice as well. Fig.…”

“…It was shown in [7] that on the square lattice M 0 < 27 6 ; a ridiculously large upper bound. On the other hand a direct computation on the Bethe lattice [8,9] with q-neighbors gives M 0 = [q/(q − 2)] 2 , which would suggest M 0 ∼ 4 for the square lattice, M 0 ∼ 3 for the cubic and M 0 ∼ 2 for the bcc lattice. Now it can be shown, using FKG inequalities, that M 0 ≥ 3 on any bipartite lattice [10], but beyond that we have no simple or convincing argument for finding M 0 .…”

“…We next present results of our simulations and refer the reader to [9] for a more detailed description of the model and additional references.…”

“…While we have no clue of how to find rigorously the actual value of M 0 or of z c (M), it was argued in [9] that for a given z and M large enough, the typical occupancy pattern on the lattice (ignoring the label I of the particles) should be like that of a one component lattice gas with nearest neighbor hard core exclusion. For the latter system Dobrushin [11] with nearest-neighbor anti-ferromagnetic interactions on the bcc lattice [13] .…”