Journal of Physical Studies 28(2), Article 2602 [19 pages] (2024) DOI: https://doi.org/10.30970/jps.28.2602 GRAND PARTITION FUNCTION FUNCTIONAL FOR SIMPLE FLUIDS I. R. Yukhnovskii , R. V. Romanik Institute for Condensed Matter Physics, NAS of Ukraine, 1, Svientsitskii St., Lviv, UA–79011, Ukraine, e-mail: romanik@icmp.lviv.ua Received 08 December 2023; accepted 12 February 2024; published online 29 May 2024

In this paper, we will systematically present the method of collective variables with a reference system for a classical many-particle interacting system in the grand canonical ensemble. The emphasis will be placed on the details of calculations. In particular, the usage of total correlation functions defined for the grand canonical ensemble allows us to investigate very accurately the cumulants of the grand partition function for the reference system. It is shown that any cumulant $\mathfrak{M}_n$ can be expressed as a product of three components: the average particle number within the reference system. Kronecker's symbol for $n$ wave vectors, and the $n$-particle structure factor.

The functional expression for the grand partition function is derived, with all coefficients explicitly defined. The coordinates of the critical point are computed in the mean field approximation.

Key words: simple fluids, collective variables, grand canonical ensemble.

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1. D. Zubarev, Dokl. Acad. Nauk SSSR 95, 757 (1954).
2. I. Yukhnovskii, Sov. Phys. JETP 7, 263 (1958).
3. I. Yukhnovskii, M. Holovko, Statistical Theory of Classical Equilibrium Systems (Naukova Dumka, Kyiv, 1980).
4. I. Yukhnovskii, Phase Transitions of the Second Order. Collective Variables Method (World Scientific, Singapore, 1987).
5. I. R. Yukhnovskii, Ukr. J. Phys.: Reviews 10, 33 (2015).
6. I. R. Yukhnovskii, Theor. Math. Phys. 79, 536 (1989);
   Crossref
7. I. Yukhnovskii, Physica A 168, 999 (1990);
   Crossref
8. I. R. Yukhnovskii, I. M. Idzyk, V. O. Kolomiets, J. Stat. Phys. 80, 405 (1995);
   Crossref
9. O. V. Patsagan, I. R. Yukhnovskii, Theor. Math. Phys. 83, 387 (1990);
   Crossref
10. I. Yukhnovskii, O. Patsahan, J. Stat. Phys. 81, 647 (1995);
    Crossref
11. A. Stuart, K. Ord, Kendall's Advanced Theory of Statistics, 6th ed. Vol. 1: Distribution Theory (John Wiley & Sons, Chichester, 2010).
12. I. Yukhnovskii, R. Romanik, preprint ICMP-23-01E (2023); https://icmp.lviv.ua/en/preprints/2023/23-01e
13. J.-P. Hansen, I. R. McDonald, Theory of Simple Liquids: with Applications to Soft Matter, 4th ed. (Academic Press, Amsterdam, 2013).
14. I. M. Idzik, V. A. Kolomiets, I. R. Yukhnovskii, Theor. Math. Phys. 73, 1204 (1987);
    Crossref
15. P. J. Cameron, Combinatorics: Topics, Techniques, Algorithms, 1st ed. (Cambridge University Press, Cambridge, 1994).
16. https://mathworld.wolfram.com/BellNumber.html
17. https://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html
18. N. W. Ashcroft, J. Lekner, Phys. Rev. 145, 83 (1966);
    Crossref
19. P. Schofield, Proc. Phys. Soc. 88), 149 (1966);
    Crossref
20. N. F. Carnahan, K. E. Starling, J. Chem. Phys. 51, 635 (1969);
    Crossref
21. I. R. Yukhnovskii, V. O. Kolomiets, I. M. Idzyk, Condens. Matter Phys. 16, 23604 (2013);
    Crossref
22. M. S. Wertheim, Phys. Rev. Lett. 10, 321 (1963);
    Crossref
23. E. Thiele, J. Chem. Phys. 39, 474 (1963);
    Crossref
24. F. H. Ree, W. G. Hoover, J. Chem. Phys. 40, 939 (1964);
    Crossref
25. I. R. Yukhnovskii, Condens. Matter Phys. 17, 43001 (2014);
    Crossref
26. P. Attard, J. Chem. Phys. 93, 7301 (1990);
    Crossref
27. Hernando, J. A. Phys. Rev. A 33, 1338–1348 Feb (1986);
    Crossref
28. S. D. M. White, Mon. Not. R. Astron. Sci. 186, 145 (1979).