ON THE STATISTICAL PROPERTIES OF SOME TRANSPORT EQUATIONS

Abstract

It is shown that, whenever a system is a linear combination of a large number of fluctuating component systems and in the "large" system a certain quantity is a constant of the motion, under equilibrium conditions that quantity is canonically distributed in the component systems and, under non-equilibrium conditions (assuming linear regression of the fluctuations and microscopic reversibility), the quantity in question is subject to a diffusivity equation with a symmetric diffusivity tensor. Application of the above to porous media hydrodynamics immediately yields the equations for miscible displacement which were laboriously obtained earlier from a special statistical model. Similarly, the corresponding equations for suspended sediment transport in rivers and for the mixing of a pollutant in air are shown to be the result of an application of the above general statement. Finally, the theory is also applied to the inertial frequency range in the statistical theory of turbulence.