Abstract:
In this paper it is shown that the limiting Gibbs distribution, whose existence was established previously by starting from the grand canonical ensemble, can also be obtained by starting from the canonical ensemble, and both distributions coincide when a certain relation exists between the parameters $\beta$ and $\mu$ (for fixed $\beta$).
The proof is based on the local limit theorem for the number of particles.
Figures: 4.
Bibliography: 12 titles.
Citation:
A. M. Khalfina, “The limiting equivalence of the canonical and grand canonical ensembles (low density case)”, Math. USSR-Sb., 9:1 (1969), 1–52
\Bibitem{Kha69}
\by A.~M.~Khalfina
\paper The limiting equivalence of the canonical and grand canonical ensembles (low density case)
\jour Math. USSR-Sb.
\yr 1969
\vol 9
\issue 1
\pages 1--52
\mathnet{http://mi.mathnet.ru/eng/sm3604}
\crossref{https://doi.org/10.1070/SM1969v009n01ABEH001281}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=264953}
Linking options:
https://www.mathnet.ru/eng/sm3604
https://doi.org/10.1070/SM1969v009n01ABEH001281
https://www.mathnet.ru/eng/sm/v122/i1/p3
This publication is cited in the following 13 articles:
Hans Zessin, Suren Poghosyan, “A central limit theorem for a classical gas”, Adv Cont Discr Mod, 2023:1 (2023)
Hans-Otto Georgii, “The equivalence of ensembles for classical systems of particles”, J Statist Phys, 80:5-6 (1995), 1341
V. A. Arzumanian, B. S. Nakhapetian, S. K. Pogosyan, “Local limit theorem for the particle number in spin lattice systems”, Theoret. and Math. Phys., 89:2 (1991), 1138–1146
K. S. Matviichuk, “Conditions of existence and stability of a solution of singular Kirkwood–Salsburg equations. Part III”, Theoret. and Math. Phys., 51:1 (1982), 372–381
V. V. Krivolapova, “Equivalence of Gibbs ensembles for classical lattice systems”, Theoret. and Math. Phys., 52:2 (1982), 803–814
O Penrose, Rep Prog Phys, 42:12 (1979), 1937
K. S. Matviichuk, “Mathematical description of the states of bose and fermi systems by the method of partial density matrices of the canonical ensemble”, Theoret. and Math. Phys., 41:3 (1979), 1067–1079
Yu. R. Dashyan, “Equivalence of the canonical and grand canonical ensembles for one-dimensional systems of quantum statistical mechanics”, Theoret. and Math. Phys., 34:3 (1978), 217–224
Yu. G. Pogorelov, “Cluster property in a classical canonical ensemble”, Theoret. and Math. Phys., 30:3 (1977), 227–232
A. M. Dolotkazina, “Local limit theorem for a system of particles without hard core”, Theoret. and Math. Phys., 27:2 (1976), 439–442
Ya. G. Sinai, “Construction of dynamics in one-dimensional systems of statistical mechanics”, Theoret. and Math. Phys., 11:2 (1972), 487–494
R. A. Minlos, A. Khaitov, “Equivalence in the limit of thermodynamic ensembles in the case of one-dimensional classical systems”, Funct. Anal. Appl., 6:4 (1972), 337–338
R. A. Minlos, A. M. Khalfina, “Two-dimensional limit theorem for the particle number and energy in
the grand canonical ensemble”, Math. USSR-Izv., 4:5 (1970), 1183–1202
Citation in format AMSBIB
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