Zwanzig-Mori equation for the time-dependent pair distribution function

Abstract

We develop a microscopic theoretical framework for the time-dependent pair distribution function starting from the Liouville equation. An exact Zwanzig-Mori equation of motion for the time-dependent pair distribution function is derived based on the projection-operator formalism. It is demonstrated that, under the Markovian approximation, our equation reduces to the so-called telegraph equation that includes the potential of mean force acting between the pair particles. With the additional approximation neglecting the inertia term, our equation takes the form of Smoluchowski's equation, which has been previously introduced with intuitive arguments and shown to satisfactorily reproduce the simulation results of the particle-pair dynamics.