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Entropy and phase transitions in partially ordered sets

Dhar, Deepak (1978) Entropy and phase transitions in partially ordered sets. Journal of Mathematical Physics, 19 (8). pp. 1711-1713. ISSN 0022-2488. doi:10.1063/1.523869.

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We define the entropy function S (ρ) =lim_(n→∞)2n^(−2)ln 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 ρ_1 and ρ_2;.083<ρ_1⩽1/4 and 3/8⩽ρ_2<48/49; such that S (ρ) has a constant value (ln2)/2 in the interval ρ_1⩽ρ⩽ρ_2; 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

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Additional Information:© 1978 American Institute of Physics. Received 16 January 1978. Online Publication Date: 11 August 2008. I would like to thank Professor Jon Mathews for many critical comments, Professor Kenneth Bogart for pointing out Ref. 1 to me, and Ms. RLou Norquist for typing the manuscript,
Issue or Number:8
Classification Code:PACS: 05.70.Fh
Record Number:CaltechAUTHORS:20120730-074232038
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Official Citation: Entropy and phase transitions in partially ordered sets Deepak Dhar J. Math. Phys. 19, 1711 (1978);
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:32779
Deposited By: Ruth Sustaita
Deposited On:30 Jul 2012 16:54
Last Modified:09 Nov 2021 21:29

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