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).

1.
For reviews see, e.g., (a) K. Huang, Statistical Mechanics (Wiley, New York, 1963);
(b) C. A. Croxton, Liquid State Physics, A Statistical Mechanical Introduction (Cambridge U.P., Cambridge, 1974), Chap. 1;
(c) G. E. Uhlenbeck and G. W. Ford, The Theory of Linear Graphs with Applications to the Theory of the Virial Development of the Properties of Gases, in Studies in Statistical Mechanics, edited by J. de Boer and G. E. Uhlenbeck (North‐Holland, Amsterdam, 1962), Vol. I;
(d) G. E. Uhlenbeck and G. W. Ford, Lectures in Statistical Mechanics (American Mathematical Society, Providence, RI, 1963).
2.
H. D.
Ursell
,
Proc. Cambridge Phil. Soc.
23
,
685
(
1927
).
3.
B.
Kahn
and
G. E.
Uhlenbeck
,
Physica
5
,
399
(
1938
).
4.
K.
Husimi
,
J. Chem. Phys.
18
,
682
(
1950
).
5.
S.
Ono
,
J. Chem. Phys.
19
,
504
(
1951
).
6.
F. Y.
Wu
,
J. Math. Phys.
4
,
1438
(
1963
);
S.
Sherman
,
J. Math. Phys.
6
,
1189
(
1965
);
G.
Stell
,
J. Math. Phys.
6
,
1193
(
1965
).
7.
J. E.
Mayer
,
J. Chem. Phys.
5
,
67
(
1937
);
J. E. Mayer and M. G. Mayer, Statistical Mechanics (Wiley, New York, 1940).
8.
J. E. Mayer, Equilibrium Statistical Mechanics (Pergamon, New York, 1968), p. 74.
9.
E. E.
Salpeter
,
Ann. Phys.
5
,
183
(
1958
);
M. S.
Green
,
J. Math. Phys.
1
,
391
(
1960
);
for a review, see, e.g., G. Stell, Cluster Expansions for Classical Systems in Equilibrium, in The Equilibrium Theory of Classical Fluids, edited by H. L. Frisch and J. L. Lebowitz (Benjamin, New York, 1964).
10.
For a review see, e.g., C. Bloch, Diagram Expansions in Quantum Statistical Mechanics, in Studies in Statistical Mechanics, edited by J. de Boer and G. E. Uhlenbeck (North‐Holland, Amsterdam, 1965), Vol. III.
11.
R.
Brout
,
Phys. Rev.
115
,
824
(
1959
).
12.
R. Brout and P. Carruthers, Lectures on the Many Electron Problem (Gordon and Breach, New York, 1969).
13.
R. Balescu, Equilibrium and Non‐Equilibrium Statistical Mechanics (Wiley, New York, 1975).
14.
R.
Kubo
,
J. Phys. Soc. Jpn.
17
,
1100
(
1962
).
15.
For more mathematical discussions of cumulants, see, e.g., H. Cramer, Random Variables and Probability Distributions, 3rd ed. (Cambridge U.P., Cambridge, 1970);
R. N. Bhattacharya and R. Ranga Rao, Normal Approximation and Asymptotic Expansions (Wiley, New York, 1976);
D. R. Brillinger, Time Series, Data Analysis and Theory, expanded edition (Holden‐Day, San Francisco, 1981).
16.
Section 2.5 of Ref. 12, and references therein.
17.
T. D.
Lee
and
C. N.
Yang
,
Phys. Rev.
113
,
1165
(
1959
);
T. D.
Lee
and
C. N.
Yang
,
116
,
25
(
1959
).,
Phys. Rev.
18.
(a) C. Berge, Graphes et Hypergraphes (Dunod, Paris, 1970) [English translation: Graphs and Hypergraphs (North‐Holland, Amsterdam, 1973)];
(b) J. E. Graver and M. E. Watkins, Combinatorics with Emphasis on the Theory of Graphs (Springer, Berlin, 1977).
19.
E.g., G. Stell, Generating Functionals and Graphs, in Graph Theory and Theoretical Physics, edited by F. Harary (Academic, New York, 1967);
or, Correlation Functions and their Generating Functionals, in Phase Transitions and Critical Phenomena, edited by C. Domb and M. S. Green (Academic, New York, 1976), Vol. 5B.
20.
A multiset is also sometimes called a selection; see, e.g., Ref. 18(b), p. 5.
21.
Equation (2.9) follows immediately from definitions (2.1):
22.
This is readily seen after replacing each exp(i) by (1+1i), as is permissible under the protection of L.
23.
In the case of an edge E, |E| is indeed the number of vertices contained in E, in our set theoretic notation. By duality, we also denote |ν| the number of edges containing the vertex v (note that one can represent a hypergraph H by its dual 18H* = (V1,V2,…,Vn), where n = |V(H)| and Vi is the set of edges containing the vertex vi).
24.
More generally, given a set of vertices W⊂V(H), we write E‐‐‐‐wE’ if there exists a path between edges E and E’ not containing any vertex of W;‐‐‐‐w is clearly an equivalence relation between edges.
25.
In detail,
where Sjf = ΠE∈SjfE.
26.
See, e.g., Ref. 1(a), Sec. 7.6 (Gibbs paradox).
27.
See, e.g., Ref. 10, p. 31.
28.
E.g., if E = k¯ = {1,2,…,k}, we have (k−1)! different cycles on k¯ corresponding to the (k−1)! distinct permutations of k−1 indices in the basic cycle (1,2,…,k), keeping one index fixed [since (i1,i2,…,ik) and a cyclic permutation of it, e.g., (ik,i1,…,ik−1,) represent the same cycle].
29.
However, there are certain cases where different irreducible blocks are‐independent, E.g., let H contain an articulation vertex v which, for some partition H = H1+H2 with V(H1)∩V(H2) = υ, is incident on only C‐edges of H1 and only/‐edges of H2. Now, C‐ and/edges, although not commuting, are not entangled by Λ in (6.15), which puts all f’s to the right of ally’s; and since furthermore all/‐edges of H. commute with all/‐edges of H2 as they have no common vertex, and likewise for C‐edges, H2 and H2 are completely de‐entangled. It follows that 〈H1f,c〉〈H2f,c and 〈Hf,cc|irr| = 0.
30.
To adapt (Bl) to the evaluation of 〈Tcx|C|, let 〈(…)〉 = 〈(…)〉.
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