Hydrodynamic equations and correlation functions (Reprinted from Annals ofPhysics, vol 24, pg 419-469, 1963)

The response of a system to an external disturbance can always be expressed in terms of time dependent correlation functions of the undisturbed system . More particularly the linear response of a system disturbed slightly from equilibrium is characterized by the expectation value in the equilibrium e nsemble. of a product of two space- and time-dependent operators. When a di sturbance leads to a very slow variation in space and time of all physical quantities, the response may alternatively be described by the linearized h ydrodynamic equations. The purpose of this paper is to exhibit the complica ted structure the correlation functions must have in order that these descr iptions coincide. From the hydrodynamic equations the slowly varying part o f the expectation values of correlations of densities of conserved quantiti es is inferred. Two illustrative examples are considered: spin diffusion an d transport is an ordinary One-component fluid. Since the descriptions are equivalent, all transport processes which occur in the nonequilibrium system must be exhibited in the equilibrium correlati on functions. Thus. when the hydrodynamic equations predict the existence o f a diffusion processes, the correlation functions will include a part whic h satisfies a diffusion equation. Similarly when sound waves occur in the n onequilibrium system, they will also be contained in the correlation functi ons. The description in terms of correlation functions leads naturally to expres sions for the transport coefficients like those discussed by Kubo. The anal ysis also leads to a number of sum rules relating the dissipative linear co efficients to thermodynamic derivatives. It elucidates the peculiarly singu lar limiting behavior these correlations must have. (C) 1963 Academic Press .