Path integrals and fluctuations in irreversible thermodynamics

We express the set of stochastic differential equations which describe fluctuations in linear irreversible thermodynamics in terms of path integrals. The stochastic terms which are added to the linearized macroscopic equations have a correlation matrix that is singular, which implies that the straightforward formulation of the problem in terms of path integrals fails. We therefore begin by constructing a path-integral representation which is valid whether or not the correlation matrix is singular. We apply this to linearized irreversible thermodynamics, but the technique is designed to be applicable to more general versions of the theory. The approach emphasizes the role of the response and correlation functions as basic elements of the theory, and we calculate these quantities explicitly for the case of density fluctuations in a fluid.