Overcomplete free energy functional for D=1 particle systems with next neighbor interactions

Details Overcomplete free energy functional for D=1 particle systems with next neighbor interactions Overcomplete free energy functional for D=1 particle systems with next neighbor interactions

2002-05-31

arxiv.org/abs/cond-mat/0205661v1

J. Stat. Phys. 111, 1125 (2003)

Physics

Condensed Matter

Soft Condensed Matter

17 pages, no figures, uses style amsart of the AMS

Scientific paper

We deduce an overcomplete free energy functional for D=1 particle systems with next neighbor interactions, where the set of redundant variables are the local block densities $\varrho_i$ of $i$ interacting particles. The idea is to analyze the decomposition of a given pure system into blocks of $i$ interacting particles by means of a mapping onto a hard rod mixture. This mapping uses the local activity of component $i$ of the mixture to control the local association of $i$ particles of the pure system. Thus it identifies the local particle density of component $i$ of the mixture with the local block density $\varrho_i$ of the given system. Consequently, our overcomplete free energy functional takes on the hard rod mixture form with the set of block densities $\varrho_i$ representing the sequence of partition functions of the local aggregates of particle numbers $i$. The system of equations for the local particle density $\varrho$ of the original system is closed via a subsidiary condition for the block densities in terms of $\varrho$. Analoguous to the uniform isothermal-isobaric technique, all our results are expressible in terms of effective pressures. We illustrate the theory with two standard examples, the adhesive interaction and the square-well potential. For the uniform case, our proof of such an overcomplete format is based on the exponential boundedness of the number of partitions of a positive integer (Hardy-Ramanujan formula) and on Varadhan's theorem on the asymptotics of a class of integrals. We also discuss the applicability of our strategy in higher dimensional space, as well as models suggested thereof.

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