Hydrodynamics and Time-Correlation Functions

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Abstract
The derivation of hydrodynamic equations, and their constitutive relations, by the time-correlation function method requires the reduction of general nonlocal equations to the usual local differential equations. This is accomplished by an approximation procedure generalizing the Chapman-Enskog method used in kinetic theory. We investigate here the validity of this method in the context of time-correlation functions characterizing hydrodynamic systems. It is pointed out that such correlation functions occurring in the nonlocal equations are not sharply peaked in space and time, as generally assumed in the literature. The procedure of expanding thermodynamic parameters and extracting their gradients from integrals over correlation functions is nevertheless shown to be valid for the systems considered. This is accomplished by demonstrating that the long-time parts of the correlation functions do not contribute, leaving the transport coefficients entirely determined by the microscopic or short-time parts. The mathematical requirements of the Chapman-Enskog procedure and the apparently contradictory analytic properties of the correlation functions for hydrodynamic systems are therefore resolved. No attempt is made to prove the assumed analytic behavior of the correlation functions from a specific force law.