Dhar, Deepak (1978) Entropy and phase transitions in partially ordered sets. Journal of Mathematical Physics, 19 (8). pp. 17111713. ISSN 00222488. http://resolver.caltech.edu/CaltechAUTHORS:20120730074232038

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Abstract
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,  
Classification Code:  PACS: 05.70.Fh  
Record Number:  CaltechAUTHORS:20120730074232038  
Persistent URL:  http://resolver.caltech.edu/CaltechAUTHORS:20120730074232038  
Official Citation:  Entropy and phase transitions in partially ordered sets Deepak Dhar J. Math. Phys. 19, 1711 (1978); http://dx.doi.org/10.1063/1.523869  
Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  32779  
Collection:  CaltechAUTHORS  
Deposited By:  Ruth Sustaita  
Deposited On:  30 Jul 2012 16:54  
Last Modified:  26 Dec 2012 15:45 
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