Entropy and phase transitions in partially ordered sets
Deepak Dhar · Journal of Mathematical Physics · 1978
We define the entropy function S (ρ) =limn→∞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.