Widom-rowlinson Model (continuum and Lattice)

Continuum Widom-Rowlinson model: In the beginning of seventies, a great progress has already been made in understanding equilibrium phase transitions and critical phenomena in classical (i.e. non-quantum) lattice models, where basic elements are localized on vertices of a regular discrete structure and interact with their nearest neighbors. The lattice gas analogy [1] of Onsager’s exact solution of the two-dimensional Ising model enabled one to understand qualitatively as well as quantitatively physical implications of the spontaneous breaking of the particle-hole symmetry on the existence and critical properties of high-density liquid and low-density vapor phases. On the other hand, little was known about the liquid-vapor equilibrium in continuum models (fluids), where the distance between interacting particles is a continuous variable. In 1970 Widom and Rowlinson (WR) introduced a simple model of the classical fluid in thermodynamic equilibrium [2], found a counterpart of the particle-hole symmetry and described implications of this symmetry on a potential liquid-vapor phase diagram. Subsequent studies proved rigorously the existence of liquid-vapor phase transitions in the WR and related models in spatial dimensions d ≥ 2; up to now WR fluids are the only continuum systems with decaying interactions for which on has this kind of rigorous results. The WR model consists of identical molecules living in an infinite d-dimensional continuous space of points r ∈ V (V → R), around the center of each molecule there is a microscopic sphere of radius σ and volume v0. The potential energy U associated with a given configuration r1, . . . , rN of N molecules is defined by U(r1, . . . , rN) = [W (r1, . . . , rN)−Nv0] ǫ/v0, (1)