PROJECTION TECHNIQUES IN NON-EQUILIBRIUM STATISTICAL MECHANICS .I. A NEW HIERARCHY OF EQUATIONS

It is shown that, using appropriate projection operators, a systematic derivation can be given of several types of kinetic equations in non equilibrium statistical mechanics. In this derivation a new hierarchy is constructed, in which functions depending on k-position and k-momentum variables are coupled to functions of the same number of position but a different number of momentum variables. Under some simplifying circumstances this hierarchy yields an equation for the one-particle distribution function f1 which agrees with that obtained by Prigogine and Balescu under the same circumstances. However, the result derived here for the two-particle distribution function f2 does not agree with the corresponding result obtained using diagrammatic methods. The conditions under which an approximate Markoffian equation can be obtained are analyzed. Hydrodynamical equations are constructed. An approximation procedure is set up for the analogue of the Boltzmann equation, patterned after the Chapman-Enskog expansion. This method can in principle be used to yield numerical results for the equations obtained. © 1969.