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Teor. Veroyatnost. i Primenen., 1964, Volume 9, Issue 4, Pages 626–643 (Mi tvp414)  

This article is cited in 13 scientific papers (total in 13 papers)

Investigation of the Conditions of the Asymptotic Existence of the Configuration Integral of the Gibbs Distribution

R. L. Dobrušin

Moscow

Abstract: Let $V$ be a cube of dimension $\nu$, with volume $|V|$. Let ${{|V|}/{N \to\nu}}$, $N\to\infty$. Let ${\mathbf x}=(x_1,\cdots ,x_N)$, $x_i\in V$, $i=1,\cdots ,N$,
$$ Q(V,N)=\int_V\dotsi\int_V\exp\{-\beta U({\mathbf x})\} dx_1…dx_N, $$
where
$$ U({\mathbf x})=\sum_{1\leqq i<j\leqq N}\Phi(|x_i-x_j|). $$
The conditions on $\Phi(y)$, which are sufficient and in some sense necessary for the existence of the finite limit
$$ \lim_{N\to\infty}\frac1N\log\frac1{{N!}}Q(V,N) $$
are given.

Full text: PDF file (973 kB)

English version:
Theory of Probability and its Applications, 1964, 9:4, 566–581

Bibliographic databases:

Received: 20.05.1964

Citation: R. L. Dobrušin, “Investigation of the Conditions of the Asymptotic Existence of the Configuration Integral of the Gibbs Distribution”, Teor. Veroyatnost. i Primenen., 9:4 (1964), 626–643; Theory Probab. Appl., 9:4 (1964), 566–581

Citation in format AMSBIB
\Bibitem{Dob64}
\by R.~L.~Dobru{\v s}in
\paper Investigation of the Conditions of the Asymptotic Existence of the Configuration Integral of the Gibbs Distribution
\jour Teor. Veroyatnost. i Primenen.
\yr 1964
\vol 9
\issue 4
\pages 626--643
\mathnet{http://mi.mathnet.ru/tvp414}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=176512}
\zmath{https://zbmath.org/?q=an:0168.23804}
\transl
\jour Theory Probab. Appl.
\yr 1964
\vol 9
\issue 4
\pages 566--581
\crossref{https://doi.org/10.1137/1109079}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. R. A. Minlos, “Lectures on statistical physics”, Russian Math. Surveys, 23:1 (1968), 137–196  mathnet  crossref  mathscinet  zmath
    2. A. M. Khalfina, “The limiting equivalence of the canonical and grand canonical ensembles (low density case)”, Math. USSR-Sb., 9:1 (1969), 1–52  mathnet  crossref  mathscinet
    3. Yu. M. Sukhov, “Odnoparametricheskaya polugruppa operatorov, porozhdennaya raspredeleniem Gibbsa”, UMN, 25:1(151) (1970), 199–200  mathnet  mathscinet  zmath
    4. R. L. Dobrushin, “Gibbsian random fields for particles without hard core”, Theoret. and Math. Phys., 4:1 (1970), 705–719  mathnet  crossref  mathscinet  zmath
    5. N. Ali, “Ustoichivost sistemy zaryazhennykh chastits”, UMN, 26:1(157) (1971), 229–230  mathnet  mathscinet
    6. I. L. Simyatitskii, “Comments on the paper “Mathematical description of the equilibrium state of classical systems on the basis of the canonical ensemble formalism” by N. N. Bogolyubov, D. Ya. Petrina, and B. I. Khatset”, Theoret. and Math. Phys., 6:2 (1971), 169–174  mathnet  crossref
    7. L. A. Pastur, “On the existence and continuity of the pressure in quantum statistical mechanics”, Theoret. and Math. Phys., 14:2 (1973), 157–163  mathnet  crossref
    8. S. S. Vallander, “A remark on the existence of the thermodynamic limit”, Theoret. and Math. Phys., 19:1 (1974), 378–380  mathnet  crossref  mathscinet
    9. S. S. Vallander, “Thermodynamic limit for many-temperature mixtures of classical neutral particles”, Theoret. and Math. Phys., 20:1 (1974), 694–703  mathnet  crossref  mathscinet
    10. Yu. G. Pogorelov, “Cluster property in a classical canonical ensemble”, Theoret. and Math. Phys., 30:3 (1977), 227–232  mathnet  crossref  mathscinet  zmath
    11. V. A. Zagrebnov, L. A. Pastur, “Singular interaction potentials in classical statistical mechanics”, Theoret. and Math. Phys., 36:3 (1978), 784–797  mathnet  crossref  mathscinet
    12. A. G. Basuev, “Theorem on the minimal specific energy for classical systems”, Theoret. and Math. Phys., 37:1 (1978), 923–926  mathnet  crossref  mathscinet
    13. A. L. Rebenko, “Mathematical foundations of equilibrium classical statistical mechanics of charged particles”, Russian Math. Surveys, 43:3 (1988), 65–116  mathnet  crossref  mathscinet  adsnasa  isi
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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