Expansions in Statistical Mechanics as Part of the Theory of Partial Differential Equations

We study perturbations of gaussian processes using a partial differential equation in (infinitely) many variables which describes what infinitesimal change in the perturbation compensates an infinitesimal change in the covariance. We derive a series representation for the solution by iterating an integral equation form of the flow equation and show that the series is majorised for short times by a corresponding solution of a Hamilton-Jacobi equation when the initial data is bounded and analytic. The resulting series solutions are generalizations of the Mayer expansion in statistical mechanics. This approach gives a remarkable identity for “connected parts” and accurate estimates which include criteria for convergence of iterated Mayer expansions.