Entropy and phase transitions in partially ordered sets

Abstract

We define the entropy function S (ρ) =lim_n→∞2n^-2ln N (n,ρ), where N (n,ρ) is the number of different partial order relations definable over a set of n distinct objects, such that of the possible n (n-1)/2 pairs of objects, a fraction ρ are comparable. Using rigorous upper and lower bounds for S (ρ), we show that there exist real numbers ρ₁ and ρ₂;.083<ρ₁≤1/4 and 3/8≤ρ₂<48/49; such that S (ρ) has a constant value (ln2)/2 in the interval ρ₁≤ρ≤ρ₂; but is strictly less than (ln2)/2 if ρ<.083 or if ρ≥48/49. We point out that the function S (ρ) may be considered to be the entropy function of an interacting "lattice gas" with long-range three-body interaction, in which case, the lattice gas undergoes a first order phase transition as a function of the "chemical activity" of the gas molecules, the value of the chemical activity at the phase transition being 1. A variational calculation suggests that the system undergoes an infinite number of first order phase transitions at larger values of the chemical activity. We conjecture that our best lower bound to S (ρ) gives the exact value of S (ρ) for all ρ.