Minimization of dimension for functional differential equations and the thermodynamics of the classical one-dimensional fluid

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The method of minimization of dimension for systems of first-order partial differential equations (PDEs) is extended analogously to systems of functional differential equations (FDEs). This method is applied to an exact first-order system of FDEs for the grand partition function of a one-dimensional classical fluid giving an alternative derivation of the pair potentials found by Baxter, for which exact thermodynamics can be obtained. These potentials satisfy a constant-coefficient ordinary differential equation (ODE). The method also gives the eigenvalue problem for the thermodynamics in these cases which is illustrated by deriving it explicitly for the simplest case, which is the exponential potential. The connection is derived between the finite system with external field to which the method applies and the infinite system without external field. This clarifies some points in Baxter's work and sheds some light on possible extensions of it.