Derivation of a Master Equation for the Relaxation of Internal Degrees of Freedom

Abstract

Using projection operator techniques and the Liouville formalism, a derivation of an exact evolution equation for the internal degrees of freedom of a molecule in a temperature bath is presented. This non‐Markovian “master equation” gives the time evolution of the diagonal part of the reduced density matrix of the internal degrees of freedom. Since the collision term in this equation depends explicitly on the intensive variables of the reservoir, the simultaneous limits of low reservoir density and long time may be used to reduce the exact equation to a Markovian master equation. Using an identity which connects the Liouville formalism to scattering theory, it is shown that the collisional transition probabilities which occur in this master equation can be written in terms of scattering cross sections. Finally, it is demonstrated that the transition probabilities satisfy the condition of detailed balance and that the master equation agrees with that obtained by the usual physical derivations.