First-order transitions from singly peaked distributions

Suppose that, in the thermodynamic limit, a single-component particle system exhibits a standard first-order transition marked by a jump in the density, ρρ, at a chemical potential μσ(T)μσ(T). In grand canonical simulations of model fluids that realize such a transition when L→∞L→∞ (where LL is the linear dimension of the simulation volume) the presence of the transition is typically signaled by the appearance of a double-peaked structure in the distribution function, PN(T,μσ;L)PN(T,μσ;L), of the particle number, NN. A simple, explicit counterexample is presented, however, that proves, contrary to popular beliefs, that the converse proposition is false: i.e., a single-peaked   distribution, PN(T,μσ;L)PN(T,μσ;L), may, when L→∞L→∞, give rise to a first-order transition. Alternatively, the existence of a first-order transition does not imply a double-peaked distribution. Systems that may exhibit such single-peaked, first-order behavior are discussed and a possible route to constructing explicit models exhibiting the phenomenon is described. Strategies to use in simulating such systems are briefly considered in the light of related studies.