  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.
