Overcomplete Free Energy Functional for D=1 Particle Systems with Next Neighbor Interactions

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 +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 +i of the given system. Consequently, our overcomplete free energy functional takes on the hard rod mixture form with the set of block densities +i representing the sequence of partition functions of the local aggregates of particle numbers i. The system of equations for the local particle density + of the original system is closed via a subsidiary condition for the block densities in terms of +. Analoguous to the uniform isothermalisobaric 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.