Phase transitions of a multicomponent Widom-Rowlinson model

Abstract

We study a multicomponent version of the ``A−B'' model of Widom and Rowlinson, generalized in a symmetric way: There is an infinite repulsive interaction between any two unlike particles. We consider both lattice and continuum versions of the model and show that the ``demixing'' transition occurs for any finite number M of components, all having the same activity. No conclusion can be drawn about this transition in the limit M→∞. It is shown, however, that another transition, in which the density is greater on one of the sublattices, appears at a finite value of M which persists for all larger M at any fixed value of the activity. In the limit M→∞, z→0, M z=ζ, const, this system apparently becomes ``equivalent'' to a one-component system with activity ζ in which there is an exclusion for occupancy of nearest neighbor sites. The latter transition then becomes the ``hard square'' transition.

The random geometry of equilibrium phases

“…As a consequence, the proof of Theorem 6.10 cannot be adapted to the case of the multitype (q ≥ 3) Widom-Rowlinson model. In fact, such a Widom-Rowlinson analogue of Theorem 6.10 is known to be false, as shown by Runnels and Lebowitz [207]; see also [57] and [184].…”

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

“…This model was first considered by Runnels and Lebowitz [7] who proved that when the number of components M is larger than some minimum M 0 then the transition from the gas phase at small values of z to the demixed phase at large values of z does not take place directly. Instead there is, at intermediate values of z, z c < z < z d , an ordered phase in which one of the sublattices (even or odd) is preferentially occupied, i.e.…”

“…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.…”

Ordering and demixing transitions in multicomponent Widom-Rowlinson models

“…existence of sharp interfaces between coexisting phases, in ν ≥ 3, at large fugacity z > z d (M), can be obtained using standard Peierls methods, see [6], [7]. A rather surprising result (at least on first sight) was found by Runnels and Lebowitz [8]. They proved that when the number of components M is larger than some minimum M 0 then the transition from the gas phase at small values of z to the demixed phase at large values of z does not take place directly.…”