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).
Skip Nav Destination
Article navigation
April 1983
Research Article|
April 01 1983
Quantum virial coefficients as cumulants of imaginary time‐ordered Mayer diagrams
Antoine Royer
Antoine Royer
Centre de Recherche de Mathématiques Appliquées, Université de Montréal, Montréal, Québec H3C 3J7, Canada
Search for other works by this author on:
J. Math. Phys. 24, 897–912 (1983)
Article history
Received:
August 18 1982
Accepted:
October 15 1982
Citation
Antoine Royer; Quantum virial coefficients as cumulants of imaginary time‐ordered Mayer diagrams. J. Math. Phys. 1 April 1983; 24 (4): 897–912. https://doi.org/10.1063/1.525779
Download citation file:
Pay-Per-View Access
$40.00
Sign In
You could not be signed in. Please check your credentials and make sure you have an active account and try again.
Citing articles via
Cascades of scales: Applications and mathematical methodologies
Luigi Delle Site, Rupert Klein, et al.
Related Content
N≥ 𝟐 symmetric superpolynomials
J. Math. Phys. (March 2017)
On some partitions of hypergraphs and cumulants having applications in statistical mechanics
J. Math. Phys. (May 1984)
Properties of Mayer cluster expansion
J. Math. Phys. (August 1982)
Mayer's Treatment of Ionic Solutions
J. Chem. Phys. (April 1957)
Maria Goeppert Mayer: Atoms, Molecules and Nuclear Shells
Physics Today (September 1986)