Abstract:
The method of collective variables is used to consider the partition function of the
three-dimensisnal Ising model. A rigorous transition is made from the phase space
of spin variables to the phase space of collective variables. It is shown that among
the set of collective variables {ρk}N there is a variable ρ0 with respect to which there is a change in the form of the distribution function on the transition through the critical point. A basis distribution with respect to the collective variables describing events at the critical point is found. In the argument of the exponential, it contains the second and fourth powers of the collective variables. To within the basis distributions, the partition function of the system is integrated over layers of the phase space of the collective variables. Recursion relations are found. The critical point is determined. The method is compared with the ε-expansion method proposed by Wilson and Fisher. The integral form of the basis distribution is used to consider the problem of block structures of the system. Besides original results, the paper collects together the results obtained by the author, Yu. K. Rudavskii, and M. P. Kozlovskii published earlier in other journals.
Citation:
I. R. Yukhnovskii, “Partition function of the three-dimensional Ising model”, TMF, 36:3 (1978), 373–399; Theoret. and Math. Phys., 36:3 (1978), 798–815
\Bibitem{Yuk78}
\by I.~R.~Yukhnovskii
\paper Partition function of the three-dimensional Ising model
\jour TMF
\yr 1978
\vol 36
\issue 3
\pages 373--399
\mathnet{http://mi.mathnet.ru/tmf3088}
\transl
\jour Theoret. and Math. Phys.
\yr 1978
\vol 36
\issue 3
\pages 798--815
\crossref{https://doi.org/10.1007/BF01035756}
Linking options:
https://www.mathnet.ru/eng/tmf3088
https://www.mathnet.ru/eng/tmf/v36/i3/p373
Erratum
Corrigendum I. R. Yukhnovskii TMF, 1979, 39:1, 143
This publication is cited in the following 21 articles:
I. R. Yukhnovskii, “Phase space of collective variables and the Zubarev transition
function”, Theoret. and Math. Phys., 194:2 (2018), 189–219
George Bokun, Dung di Caprio, Myroslav Holovko, Vyacheslav Vikhrenko, “The system of mobile ions in lattice models: Screening effects, thermodynamic and electrophysical properties”, Journal of Molecular Liquids, 270 (2018), 183
І.R. Yukhnovskii, M.P. Kozlovskii, І.V. Pilyuk, “Metod rozrakhunku vіlnoï energіï trivimіrnoï іzingopodіbnoï sistemi z vrakhuvannyam popravki na userednennya potentsіalu vzaєmodіï”, Ukr. J. Phys., 57:1 (2012), 80
Kozlovskii, MP, “Microscopic description of the critical behavior of three-dimensional Ising-like systems in an external field”, Physical Review B, 73:17 (2006), 174406
Yukhnovskii, IR, “Study of the critical behaviour of three-dimensional Ising-like systems on the basis of the rho(6) model with allowance for microscopic parameters: I. High-temperature region”, Journal of Physics-Condensed Matter, 14:43 (2002), 10113
Yukhnovskii, IR, “Thermodynamics of three-dimensional Ising-like systems in the higher non-Gaussian approximation: Calculational method and dependence on microscopic parameters”, Physical Review B, 66:13 (2002), 134410
Z. E. Usatenko, M. P. Kozlovskii, “Thermodynamic characteristics of the classicaln-vector magnetic model in three dimensions”, Phys. Rev. B, 62:14 (2000), 9599
Pylyuk, IV, “Description of critical behavior of Ising ferromagnet in the rho(6) model approximation taking into account confluent correction. I. Region above the phase transition point”, Low Temperature Physics, 25:11 (1999), 877
I. V. Pylyuk, “Critical behavior of the three-dimensional Ising sistem: Dependence of themodynamic characteristics on microscopic parameters”, Theoret. and Math. Phys., 117:3 (1998), 1459–1482
M.P. Kozlovskii, I.V. Pylyuk, V.V. Dukhovii, “Equation of state of the 3D Ising model with an exponentially decreasing potential in the external field”, Journal of Magnetism and Magnetic Materials, 169:3 (1997), 335
V. V. Dukhovyi, M. P. Kozlovskii, I. V. Pylyuk, “Equation of state in 3-D Ising model from microscopic level calculation”, Theoret. and Math. Phys., 107:2 (1996), 650–666
M. P. Kozlovskii, I. V. Pylyuk, “Entropy and specific heat of the 3D ising model as functions of temperature and microscopic parameters of the system”, Physica Status Solidi (b), 183:1 (1994), 243
M. P. Kozlovskii, I. V. Pylyuk, I. R. Yukhnovskii, “Thermodynamic functions of three-dimensional ising model near the phase transition point with allowance for corrections to scaling. I. The case $T>T_c$”, Theoret. and Math. Phys., 87:2 (1991), 540–556
M. P. Kozlovskii, I. V. Pylyuk, I. R. Yukhnovskii, “Thermodynamic functions of three-dimensional ising model near the phase transition point with allowance for corrections to scaling. II. The case $T<T_c$”, Theoret. and Math. Phys., 87:3 (1991), 641–656
M. P. Kozlovskii, “Nonasymptotic form of the recursion relations of the three-dimensional Ising model”, Theoret. and Math. Phys., 78:3 (1989), 300–308
N.S. Gonchar, “Correlation functions of some continuous model systems and description of phase transitions”, Physics Reports, 172:5 (1989), 175
Yu. V. Kozitskii, “Hierarchical vector model of a ferromagnet in the method of collective variables. The Lee–Yang theorem”, Theoret. and Math. Phys., 58:1 (1984), 63–71
Yu. V. Kozitskii, I. R. Yukhnovskii, “Generalized hierarchical model of a scalar ferromagnet in the method of collective variables”, Theoret. and Math. Phys., 51:2 (1982), 490–497
V. A. Onischuk, “Collective variables. Correlation functions on Gaussian functionals”, Theoret. and Math. Phys., 51:3 (1982), 582–593
I. A. Vakarchuk, Yu. K. Rudavskii, I. R. Yukhnovskii, “Approximate renormalization group transformation in the theory of phase transitions. I. Differential equation of the renormalization group”, Theoret. and Math. Phys., 50:2 (1982), 204–209
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