Theoretical and Mathematical Physics

, Volume 194, Issue 3, pp 390–403 | Cite as

Obtaining the Thermodynamic Relations for the Gibbs Ensemble Using the Maximum Entropy Method

  • V. V. Ryazanov


As a generating functional of the Gibbs ensemble, we use the Laplace transform of the complex (or generalized) Poisson measure. We use the maximum entropy principle to determine the form of the generating function of this distribution. We consider the cases where only the mathematical expectation is known and where the mathematical expectation and the second moment are known. In the latter case, the equation of state has a transcendental form. In the both cases, if there is no interaction, then the obtained relations lead to expressions for an ideal gas.


Gibbs system grand canonical ensemble generalized Poisson distribution maximum entropy principle 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    I. I. Ivanchik, “Analytic representation for the equation of state in classical statistical mechanics,” Theor. Math. Phys., 108, 958–976 (1996).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    E. T. Jaynes, “Information theory and statistical mechanics. I,” Phys. Rev., 106, 620–630 (1957); “Information theory and statistical mechanics. II,” Phys. Rev., 108, 171–190 (1957).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    P. Harremoës and F. Topsøe, “Maximum entropy fundamentals,” Entropy, 3, 191–226 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 3a.
    L. M. Martyushev and V. D. Seleznev, “Maximum entropy production principle in physics, chemistry, and biology,” Phys. Rep., 426, 1–45 (2006).ADSMathSciNetCrossRefGoogle Scholar
  5. 4.
    N. N. Bogoliubov, Problems of Dynamical Theory in Statistical Physics [in Russian], Gostekhizdat, Moscow (1946).Google Scholar
  6. 5.
    N. N. Bogolyubov Jr. and A. K. Prikarpatskii, “Bogolyubov generating functional method in statistical mechanics and the analog of the transformation to collective variables,” Theor. Math. Phys., 66, 305–317 (1986).CrossRefGoogle Scholar
  7. 6.
    B. V. Moshchinskii, C. Rodriguez, and V. K. Fedyanin, “Generating functional and the functional analog of Bogolyubov’s variational method,” Theor. Math. Phys., 45, 1010–1016 (1980).CrossRefGoogle Scholar
  8. 7.
    G. I. Nazin, “Method of the generating functional,” J. Soviet Math., 31, 2859–2886 (1985).CrossRefzbMATHGoogle Scholar
  9. 8.
    G. I. Nazin, “Description of Gibbs random fields by the generating functional method,” Theor. Math. Phys., 42, 251–257 (1980).MathSciNetCrossRefGoogle Scholar
  10. 9.
    V. V. Krivolapova and G. I. Nazin, “Generating functional method and Gibbs random fields on countable sets,” Theor. Math. Phys., 47, 514–532 (1981).CrossRefGoogle Scholar
  11. 10.
    V. V. Ryazanov, “Construction of correlation functions of a complex statistical system [in Russian],” Izv. Vuzov SSSR Fizika, 11, 129–130 (1978).Google Scholar
  12. 11.
    V. V. Ryazanov, “Models of a generating functional of a statistical system in the phase transition region [in Russian],” Izv. Vuzov SSSR Fizika, 9, 44–47 (1983).Google Scholar
  13. 12.
    B. A. Sevast’yanov, Branching Processes [in Russian], Nauka, Moscow (1971).zbMATHGoogle Scholar
  14. 13.
    I. N. Kovalenko, N. Yu. Kuznetsov, and V. M. Shurenkov, Random Processes: Handbook [in Russian], Naukova Dumka, Kiev (1983).zbMATHGoogle Scholar
  15. 14.
    V. V. Ryazanov, “Modeling of the equation of state of a supersaturated vapor and the dependence of its equilibrium pressure on the radius of the germinal drop [in Russian],” Zhurn. Fiz. Khimii, 58, No. 1, 72–74 (1984).Google Scholar
  16. 15.
    V. V. Ryazanov, “Obtaining different equations of state of a substance by specifying the Levy dimension of the generalized Poisson-distribution,” High Temperature, 21, 834–839 (1983).Google Scholar
  17. 16.
    V. V. Ryazanov, “Simulation of the generating function for the number of particles by a branching process with immigration [in Russian],” in: Physics of the Liquid State, Vol. 11, Vishcha Shkola, Kiev (1983), pp. 40–44; “Modeling of thermodynamic properties of a Gibbs statistical system [in Russian],” in: Physics of the Liquid State, Vol. 17, Vishcha Shkola, Kiev (1989), pp. 28–41; “A constructive description of pure substances and mixtures by relations of the type of the van der Waals equation [in Russian],” in: Physics of the Liquid State, Vol. 18, Vishcha Shkola, Kiev (1990), pp. 5–14; “Analytic modeling of Gibbs systems [in Russian],” in: Physics of the Liquid State, Vol. 19, Vishcha Shkola, Kiev (1991), pp. 24–35Google Scholar
  18. 16a.
    V. V. Ryazanov and O. K. Zakusilo, “Description of statistical systems by means of a random process of death and immigration [in Russian],” in: Physics of the Liquid State, Vol. 12, Vishcha Shkola, Kiev (1984), pp. 85–93.Google Scholar
  19. 17.
    R. L. Dobrushin, “On the Poisson law for the distribution of particles in space [in Russian],” Ukrain. Matem. Zhurn., 8, No. 2, 127–134 (1956).Google Scholar
  20. 18.
    R. L. Dobrushin and Yu. M. Sukhov, “Asymptotic behavior for some degenerate models of the time evolution of systems with an infinite number of particles,” J. Soviet Math., 16, 1277–1340 (1981).CrossRefzbMATHGoogle Scholar
  21. 19.
    Yu. M. Sukhov, “Convergence to a Poisson distribution for certain models of particle motion,” Math. USSR-Izv., 20, 137–155 (1983).CrossRefzbMATHGoogle Scholar
  22. 20.
    A. L. Rebenko, “Poisson measure representation and cluster expansion in classical mechanics,” Commun. Math. Phys., 151, 427–435 (1993).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 21.
    D. J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes, Vol. 1, Elementary Theory and Methods, Springer, New York (2002).Google Scholar
  24. 22.
    R. A. Minlos, “Lectures on statistical physics,” Russ. Math. Surveys, 23, 137–196 (1968).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 23.
    A. L. Rebenko and V. A. Bolukh, “Infinite-dimensional analysis and statistical mechanics [in Ukrainian],” Zbirnik Prats’ In-tu Matematiki NAN Ukraini, 11, 281–339 (2014).zbMATHGoogle Scholar
  26. 24.
    D. L. Finkelshtein, Y. G. Kondratiev, and M. J. Oliveira, “Glauber dynamics in the continuum via generating functionals evolution,” Complex Anal. Oper. Theory, 6, 923–945 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  27. 25.
    V. A. Boluh and O. L. Rebenko, “An exponential representation for some integrals with respect to Lebesgue–Poisson measure,” Methods Funct. Anal. Topol., 20, 186–192 (2014).MathSciNetzbMATHGoogle Scholar
  28. 26.
    D. Finkel’shtein, “Stochastic dynamics of continuous systems [in Ukrainian],” Mizhdistsiplinarni Doslidzhennya Skladnikh Sistem, 6, 5–48 (2015).Google Scholar
  29. 27.
    O. E. Lanford III, “Time evolution of large classical systems,” in: Dynamical Systems, Theory, and Applications (Lect. Notes Phys., Vol. 38, J. Moser, ed.), Springer, Berlin, Heidelberg (1975), pp. 1–111.Google Scholar
  30. 28.
    D. Ruelle, “Superstable interactions in classical statistical mechanics,” Commun. Math. Phys., 18, 127–159 (1970).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. 29.
    A. L. Rebenko, “Cell gas model of classical statistical systems,” Rev. Math. Phys., 25, 1330006 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  32. 30.
    A. L. Rebenko and M. V. Tertychnyi, “On stability, superstability, and strong superstability of classical systems of statistical mechanics,” Meth. Funct. Anal. Topology, 14, 287–296 (2008).MathSciNetzbMATHGoogle Scholar
  33. 31.
    C. J. Preston, Gibbs States on Countable Sets (Cambridge Tracts Math., Vol. 68), Cambridge University Press, London (1974).CrossRefzbMATHGoogle Scholar
  34. 32.
    H.-O. Georgii, Canonical Gibbs Measure (Lect. Notes Math. Phys., Vol. 760), Springer, Berlin (1979); Gibbs Measures and Phase Transitions (De Gruyter Stud. Math., Vol. 9), Walter de Gruyter, Berlin (1988).Google Scholar
  35. 33.
    T. L. Hill, Statistical Mechanics: Principles and Selected Applications, McGraw-Hill, New York (1956).zbMATHGoogle Scholar
  36. 34.
    J. E. Mayer and M. Goeppert-Mayer, Statistical Mechanics, Wiley, New York (1940).zbMATHGoogle Scholar
  37. 35.
    S. Albeverio, Yu. G. Kondratiev, and M. Röckner, “Analysis and geometry on configuration spaces: The Gibbsian case,” J. Funct. Anal., 157, 242–291 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
  38. 36.
    Y. G. Kondratiev, J. L. Silva, and L. Streit, “Differential geometry on compound Poisson space,” Methods Funct. Anal. Topology, 4, 32–58 (1998); arXiv:math/9908059v1 (1999).MathSciNetzbMATHGoogle Scholar
  39. 37.
    A. M. Vershik, I. M. Gel’fand, and M. I. Graev, “Representations of the group of diffeomorphisms,” Russ. Math. Surveys, 30, No. 6, 1–50 (1975).MathSciNetzbMATHGoogle Scholar
  40. 38.
    V. I. Tikhonov and M. A. Mironov, Markov Processes [in Russian], Sov. Radio, Moscow (1977).zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Institute for Nuclear ResearchNational Academy of Sciences of UkraineKievUkraine

Personalised recommendations