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

Abstract

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.