Quantum Virial Expansion

The usual quantum mechanical cluster expansion in the theory of imperfect gases involves the trace of operators of the form $\exp (-\frac{H_n}{\mathrm{kT}})$, where $H_n$ is the $n$-body Hamiltonian. It is shown here that this expansion is identical with the recently derived cluster expansion in terms of the collision lifetime matrices ${\mathrm{Q}}_n\mathrm{}\left(E\right)\mathrm{}$, in which the characteristic terms involve the expressions $\mathrm{Tr}\left[\exp \right(-\frac{E}{\mathrm{kT}}\left){\mathrm{Q}}_n\mathrm{}\right(E\left)\right]$.