Calculation of mode lifetimes in weakly anharmonic solids using self-consistent ensemble eigenstates of the Liouvillian

Recent schemes for the efficient calculation of vibrational mode frequencies and lifetimes in weakly anharmonic systems such as solid crystal structures have shown promise, with simple relationships being observed between low order moments of the Liouvillian and the observed lifetime. While only shown empirically, two parameters seemed sufficient to determine the lifetime to high accuracy over a large range of temperatures for a few simple low degree of freedom models and a Lennard–Jones solid. In this paper we analytically reproduce the numerically observed relationship using a few generally applicable assumptions about the density of states. This advance is made by studying the autocorrelation based on an “ensemble eigenstate” of the Liouvillian. We propose the use of this ensemble eigenstate as the basis for autocorrelations to simplify calculations in richer systems. An analysis is also performed of the relationship between canonical and microcanonical autocorrelation functions and a long standing error in the long time behavior of such functions originally presented by Sen, Sinkovits, and Chakravarti is corrected.