Quantum virial coefficients as cumulants of imaginary time-ordered Mayer diagrams

Abstract

The virial coefficients for a quantum gas (including quantum statistics) are expressed as sums of cumulants of connected (generalized) Mayer diagrams, the cumulants being built on the irreducible blocks of the diagrams. The Mayer diagrams are defined for the quantum case in terms of imaginary time-ordered exponentials, the quantum statistics being incorporated in the guise of multiparticle interactions. In order to extend Mayer diagrams to multiparticle interactions, we utilize terminology and methods from the theory of hypergraphs. The virial coefficients naturally separate into a quantum Boltzmann gas contribution, an ideal quantum gas contribution, and a final term expressing correlations between dynamics and statistics. In the classical limit, connected Mayer diagrams factorize into their irreducible blocks; the cumulants over irreducible blocks then vanish (by a basic property of cumulants), except for diagrams which are themselves irreducible, whence the classical result of Mayer (extended to multiparticle interactions). In the quantum case, the imaginary time ordering prevents the factorization into irreducible blocks by time entangling them. As a further illustration of the use of hypergraph-cumulant methods, we directly deduce the expressions of the virial coefficients in terms of Ursell–Kahn–Uhlenbeck cluster functions (the ideal quantum gas contribution naturally appears in that form).