Entropy and phase transitions in partially ordered sets
نویسنده
چکیده
We define the entropy function S(p) = lim.-~zn -zln N(n ,p), where N(n ,p) is the number of different partial order relations definable over a set of n distinct objects, such that of the possible n ( n I )/2 pairs of objects, a fraction p are comparable. Using rigorous upper and lower bounds for S(p), we show that there exist real numbers p1 and p2; .083 < p1 ~ 1/4 and 3/8 ~ p2 < 48/49; such that S(p) has a constant value (Jn2)/2 in the interval p1 ~ p s p2; but is strictly Jess than (ln2)/2 if p ~ .083 or if p ~ 48/49. We point out that the function S(p) 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 I. 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(p) gives the exact value of S(p) for all p.
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