Relation between Time Averages and Ensemble Averages in the Statistical Dynamics of Continuous Media

The method of correlations, which has been widely used in the theory of turbulence, is applied to the case of a field that satisfies linear equations. It is found that most of the difficulties encountered in the theory of turbulence disappear when the equations are linear and that a number of general results can be obtained.In particular, it is possible to use an analog of Gibbs' procedure of replacing a time average by an ensemble average. The mathematical foundations of this procedure are discussed and shown to be considerably different than for dynamical systems with a finite number of degrees of freedom, to which the method is usually applied. In particular, it is shown that it can be applied to dissipative fields excited by a randomly varying force.Three examples are treated in some detail: statistically uniform but nonisotropic fields; statistically nonuniform, dissipative fields excited by a random force; and the scattering of monochromatic radiation by inhomogeneities of the refractive index.The method of ensembles, as applied to these problems, appears to have a mathematical foundation that is rather simpler than in other cases that have been considered in the past, but several basic and unsolved problems are mentioned.