Bethe-lattice calculations for the phase diagram of a two-state Janus   gas
  We use a simple lattice statistical model to analyze the effects of directional interactions on the phase diagram of a fluid of two-state Janus particles. The problem is formulated in terms of nonlinear recursion relations along the branches of a Cayley tree. Directional interactions are taken into account by the geometry of this graph. Physical solutions on the Bethe lattice (the deep interior of a Cayley tree) come from the analysis of the attractors of the recursion relations. We investigate a number of situations, depending on the concentrations of the types of Janus particles and the parameters of the potential, and make contact with results from recent numerical simulations. 
Danilo B. Liarte, and Silvio R. Salinas
cond-mat.soft cond-mat.stat-mech