To the kinetic theory of dense gases and liquids. Calculation of quasi-equilibrium particle distribution functions by the method of collective variables

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dc.citation.epage458
dc.citation.issue2
dc.citation.journalTitleМатематичне моделювання та комп'ютинг
dc.citation.spage440
dc.contributor.affiliationНаціональний університет “Львівська політехніка”
dc.contributor.affiliationІнститут фізики конденсованих систем НАН України
dc.contributor.affiliationLviv Polytechnic National University
dc.contributor.affiliationInstitute for Condensed Matter Physics of the National Academy of Sciences of Ukraine
dc.contributor.authorТокарчук, Михайло Васильович
dc.contributor.authorTokarchuk, M. V.
dc.coverage.placenameЛьвів
dc.coverage.placenameLviv
dc.date.accessioned2025-03-04T11:14:26Z
dc.date.created2022-02-28
dc.date.issued2022-02-28
dc.description.abstractНа основі ланцюжка рівнянь ББГКІ з модифікованою граничною умовою, що враховує багаточастинкові кореляції, отримано кінетичні рівняння у наближенні “парних” зіткнень та поляризаційному наближенні, тобто з врахуванням взаємодії через третю частинку. При цьому враховувалась специфіка модельного подання парного потенціалу взаємодії частинок через короткосяжну та далекосяжну частини. У випадку короткосяжного потенціалу у вигляді потенціалу твердих сфер отримано вклад ревізованої теорії Енскога у повний інтеграл зіткнення кінетичного рівняння. В інтеграли зіткнень входять парні квазірівноважні функції розподілу, які залежать від нерівноважних середніх значень густини кількості частинок та оберненої температури. Застосовано метод колективних змінних Юхновського для розрахунку парної квазірівноважної функції розподілу з виділенням короткосяжної та далекосяжної частин у потенціалі взаємодії частинок. При цьому система із короткодіючою взаємодією розглядається як система відліку.
dc.description.abstractBased on a chain of BBGKI equations with a modified boundary condition that takes into account multiparticle correlations, kinetic equations in the approximate "pairs" collisions and in the polarization approximation, taking into account the interaction through the third particle, obtained. The specifics of the model representation of the pair potential of particle interaction through short-range and long-range parts were taken into account. In the case of the short-range potential in the form of the potential of solid spheres, the contribution of Enskog's revised theory to the complete integration of the collision of the kinetic equation is obtained. The collision integrals include paired quasi-equilibrium distribution functions that depend on the nonequilibrium mean values of the particle number density and the inverse temperature. The method of collective variables Yukhnovskii is applied for the calculation of pair quasi-equilibrium distribution function with an allocation of short-range and long-range parts in the potential of the interaction of particles. In this case, the system with short-range interaction is considered as a frame of reference.
dc.format.extent440-458
dc.format.pages19
dc.identifier.citationTokarchuk M. V. To the kinetic theory of dense gases and liquids. Calculation of quasi-equilibrium particle distribution functions by the method of collective variables / M. V. Tokarchuk // Mathematical Modeling and Computing. — Lviv : Lviv Politechnic Publishing House, 2022. — Vol 9. — No 2. — P. 440–458.
dc.identifier.citationenTokarchuk M. V. To the kinetic theory of dense gases and liquids. Calculation of quasi-equilibrium particle distribution functions by the method of collective variables / M. V. Tokarchuk // Mathematical Modeling and Computing. — Lviv : Lviv Politechnic Publishing House, 2022. — Vol 9. — No 2. — P. 440–458.
dc.identifier.doidoi.org/10.23939/mmc2022.02.440
dc.identifier.urihttps://ena.lpnu.ua/handle/ntb/63444
dc.language.isoen
dc.publisherВидавництво Львівської політехніки
dc.publisherLviv Politechnic Publishing House
dc.relation.ispartofМатематичне моделювання та комп'ютинг, 2 (9), 2022
dc.relation.ispartofMathematical Modeling and Computing, 2 (9), 2022
dc.relation.references[1] Van der Waals J. D. On the Continuity of the Gas and Liquid State. Ph.D. Thesis, University of Leiden, Leiden (1873).
dc.relation.references[2] Zwanzig R. W. High Temperature Equation of State by a Perturbation Method. I. Nonpolar Gases. J. Chem. Phys. 22 (8), 1420–1426 (1954).
dc.relation.references[3] Rowlinson J. S., Widom B. Molecular Theory of Capillarity. Clarendon, Oxford, Chap. 1.6. (1982).
dc.relation.references[4] Widom B. Intermolecular Forces and the Nature of the Liquid State. Science. 157 (3787), 375–382 (1967).
dc.relation.references[5] Fisher M. E. Correlation Functions and the Critical Region of Simple Fluids. J. Math. Phys. 5, 944–962 (1964).
dc.relation.references[6] Widom B. Equation of State in the Neighborhood of the Critical Point. J. Chem. Phys. 43, 3898–3905 (1965).
dc.relation.references[7] Longuet-Higgens H. C., Widom B. A rigid sphere model for the melting of argon. Mol. Phys. 8 (6), 549–556 (1964).
dc.relation.references[8] Barker J. A., Henderson D. Perturbation Theory and Equation of State for Fluids. II. A Successful Theory of Liquids. J. Chem. Phys. 47, 4714–4721 (1967).
dc.relation.references[9] Weeks J. D., Chandler D., Andersen H. C. Role of Repulsive Forces in Determining the Equilibrium Structure of Simple Liquids. J. Chem. Phys. 54, 5237–5247 (1971).
dc.relation.references[10] Kushickt J., Berne B. J. Role of attractive forces in self-diffusion in dense Lennard–Jones fluids. J. Chem. Phys. 59, 3732–3736 (1973).
dc.relation.references[11] Chandler D. Equilibrium Structure and Molecular Motion in Liquids. Acc. Chem. Res. 7, 246–251 (1974).
dc.relation.references[12] Barker J. A., Henderson D. What is “liquid”? Understanding the states of matter. Reviews of Modern Physics. 48 (4), 587–671 (1976).
dc.relation.references[13] Dean D. P., Kushick J. N. On the role of attractive forces in the cage effect in simple liquids. J. Chem. Phys. 76 (1), 619–621 (1982).
dc.relation.references[14] Sung W., Dahler J. S. Theory of transport processes in dense fluids. J. Chem Phys. 80 (7), 3025–3037 (1984).
dc.relation.references[15] Barrat J. L., Hansen J. P. Basic Concepts for Simple and Complex Liquids. Cambridge University Press, Cambridge (2003).
dc.relation.references[16] Hansen J. P., McDonald J. R. Theory of Simple Liquids. Academic, New York (2005).
dc.relation.references[17] Chandler D. Interfaces and the driving force of hydrophobic assembly. Nature. 437, 640–646 (2005).
dc.relation.references[18] Kirchner B. Theory of Complicated Liquids: Investigation of Liquids, Solvents and Solvent Effects with Modern Theoretical Methods. Phys. Rep. 440 (1–3), 1–111 (2007).
dc.relation.references[19] Berthier L., Tarjus G. Nonperturbative Effect of Attractive Forces in Viscous Liquids. Phys. Rev. Lett. 103, 170601 (2009).
dc.relation.references[20] Melnyk R., Nezbeda I., Henderson D., Trokhymchuk A. On the role of the reference system in perturbation theory: An augmented van der Waals theory of simple fluids. Fluid Phase Equilibria. 279 (1), 1–10 (2009).
dc.relation.references[21] Pedersen U. R., Schroder T. B., Dyre J. C. Repulsive Reference Potential Reproducing the Dynamics of a Liquid with Attractions. Phys. Rev. Lett. 105 (15), 157801 (2010).
dc.relation.references[22] Toxvaerd S., Dyre J. C. Communication: Shifted forces in molecular dynamics. J. Chem. Phys. 134 (8), 081102 (2011).
dc.relation.references[23] Toxvaerd S., Dyre J. C. Role of the first coordination shell in determining the equilibrium structure and dynamics of simple liquids. J. Chem. Phys. 135 (13), 134501 (2011).
dc.relation.references[24] Berthier L., Tarjus G. The role of attractive forces in viscous liquids. J. Chem. Phys. 134 (21), 214503 (2011).
dc.relation.references[25] Ingebrigtsen T. S., Schrøder T. B., Dyre J. C. What Is a Simple Liquid? Phys. Rev. X. 2 (1), 011011 (2012).
dc.relation.references[26] Hordiichuk V., Trokhymchuk A., Skvara J., Nezbed I. Excluded volume interaction in a Lennard–Jones fluid. Preprint of the Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, ICMP-18-07E (2018).
dc.relation.references[27] Mryglod I. M., Omelyan I. P., Tokarchuk M. V. Generalized Collective Modes for the Lennard–Jones fluid. Mol. Phys. 84 (2), 235–259 (1995).
dc.relation.references[28] Mryglod I. M., Folk R., Tokarchuk M. V. On the hydrodynamic theory of a magnetic liquid. I. General description. Physica A. 220 (3–4), 325–348 (1995).
dc.relation.references[29] Mryglod I. M., Omelyan I. P., Tokarchuk M. V. Wavevector- and frequency-dependent shear viscosity of water: the modified collective mode approach and molecular dynamics calculations. Condens. Matter Phys. 8 (1), 25–46 (2005).
dc.relation.references[30] Markiv B., Omelyan I., Tokarchuk M. Consistent description of kinetics and hydrodynamics of weakly nonequilibrium processes in simple liquids. J. Stat. Phys. 155, 843–866 (2014).
dc.relation.references[31] Bogolubov N. N. Problems of a dynamical theory in statistical physics. In: Studies in statistical mechanics. Vol. 1. Amsterdam, North-Holland (1962).
dc.relation.references[32] Chapman S., Cowling T. The mathematical theory of non-uniform gases: an account of the kinetic theory of viscosity, thermal conduction and diffusion in gases. Cambridge, University Press, 3rd ed. (1970).
dc.relation.references[33] Ferziger J. H., Kaper H. G. Mathematical Theory of Transport Processes in Gases. North-Holland, Amsterdam (1972).
dc.relation.references[34] Resibois P., de Leener M. Classical Kinetic Theory of Fluids. New York, John Willey and Sons (1977).
dc.relation.references[35] Humenyuk Y. A., Tokarchuk M. V. Extension of Hydrodynamic Balance Equations for Simple Fluids. J. Stat. Phys. 142, 1052–1084 (2011).
dc.relation.references[36] Enskog D. Kinetische Theorie der Warmeleitung Reibung und Selbstdiffusion in gevissen verdichtetten Gasen und Flussigkeiten. Svenska Vetenskapsaked. Handl. 63, 4 (1922).
dc.relation.references[37] Van Beijeren H., Ernst M. H. The modified Enskog equation. Physica. 68 (3), 437–456 (1973).
dc.relation.references[38] Van Beijeren H., Ernst M. H. The modified Enskog equation for mixtures. Physica. 70 (2), 225–242 (1973).
dc.relation.references[39] Polewczak J., Stell G. Transport Coefficients in Some Stochastic Models of the Revised Enskog Equation. J. Stat. Phys. 109, 569–590 (2002).
dc.relation.references[40] Polewczak J. A review of the kinetic modelings for non-reactive and reactive dense fluids: Questions and problems. Rivista di Matematica della Universita’ di Parma. 4, 23–55 (2001).
dc.relation.references[41] Polewczak J. Hard-sphere kinetic models for inert and reactive mixtures. J. Phys.: Condens. Matter. 28, 414022 (2016).
dc.relation.references[42] Davis H. T., Rice S. A., Sengers J. V. On the kinetic theory of dense fluids. IX. The fluid of rigid spheres with a square-well attraction. J. Chem. Phys. 35 (6), 2210–2233 (1961).
dc.relation.references[43] Karkheck J., van Beijeren H., de Schepper I., Stell G. Kinetic theory and H theorem for a dense squarewell fluid. Phys. Rev. A. 32 (4), 2517–2520 (1985).
dc.relation.references[44] Wilbertz H., Micheis J., van Beijeren H., Leegwater J. A. Self-Diffusion of Particles Interacting Through a Square-Well or Square-Shoulder Potential. J. Stat. Phys. 53 (5/6), 1155–1177 (1988).
dc.relation.references[45] Van Beijeren H., Karkheck J., Sengers J. V. Nonequilibrium temperature and bulk viscosity for a dense fluid of square-well molecules. Phys. Rev. A. 37 (6), 2247–2250 (1988).
dc.relation.references[46] Leegwater J. A., van Beijeren H., Michels P. J. Linear kinetic theory of the square-well fliud. J. Phys.: Condens. Matter. 1 (1), 237–255 (1989).
dc.relation.references[47] Dufty J. W., Mo K. C., Gubbins K. E. Models for self-diffusion in the square well fluid. J. Chem. Phys. 94 (4), 3132–3140 (1991).
dc.relation.references[48] Garland G. E., Dufty J. W. Bound state contributions to transport coefficients. J. Chem. Phys. 95 (4), 2702–2716 (1991).
dc.relation.references[49] Yu Y.-X., Han M.-H., Gao G.-H. Self-di†usion in a Nuid of square-well spheres. Phys. Chem. Chem. Phys. 3, 437–443 (2001).
dc.relation.references[50] Dwivedee D. K., Srivastava R., Khanna K. N. Transport coefficients of square-well fluids. Fluid Phase Equilibria. 263 (2), 199–204 (2008).
dc.relation.references[51] Takada S., Saitoh K., Hayakawa H. Kinetic theory for dilute cohesive granular gases with a square-well potential. Phys. Rev. E. 94 (1), 012906 (2016).
dc.relation.references[52] Tokarchuk M. V., Omelyan I. P. To the kinetic theory of processes transfer in dense gases. Preprint of the Institute for Theoretical Physics of the Academy of Sciences of Ukraine, ITР-87-152R (1987), (in Russian).
dc.relation.references[53] Tokarchuk M. V., Omelyan I. P. The kinetic equation for the systems with multistage interparticle potential. H-theorem. Preprint of the Institute for Theoretical Physics of the Academy of Sciences of Ukraine, ITР-89-49R (1989).
dc.relation.references[54] Tokarchuk M. V., Omelyan I. P. Model kinetic equations for dense gases and liquids. Ukr. Phys. J. 35 (8), 1255–1261 (1990).
dc.relation.references[55] Omelyan I. P., Tokarchuk M. V. Kinetic equation for liquids with a multistep potential of interaction. H-theorem. Physica A. 234 (1–2), 89–107 (1996).
dc.relation.references[56] Tokarchuk M. V., Omelyan I. P., Kobryn O. E. Kinetic equation for liquids ith a multistep potential of interaction: Calculation of transport coefficient. Phys. Rev. E. 62 (6), 8021–8036 (2000).
dc.relation.references[57] Tokarchuk M. V., Humenyuk Y. A. Hydrodynamic equations for dense fluid mixtures with multistep interaction between particles. Condens. Matter Phys. 10 (2), 151–163 (2007).
dc.relation.references[58] Tokarchuk M. V., Humenyuk Y. A. Derivation of hydrodynamic equations for dense gaseous mixtures with multi-step interaction between particles. Preprint of the Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, IСМР-04-09U (2004), (in Ukrainian).
dc.relation.references[59] Humenyuk Y. A., Tokarchuk M. V. Limit behavior of kinetic theory for a system with multistage potential. Ukr. Phys. J. 55 (4), 453–460 (1999), (in Ukrainian).
dc.relation.references[60] Rice S. A., Allnatt A. R. On the kinetic theory of dense fluids. VI. Singlet distribution function for rigid spheres with an attractive potential. J. Chem. Phys. 34 (6), 2144–2155 (1961).
dc.relation.references[61] Vogelsang R., Hoheisel C. Rice-Allnatt theory: Comparison of calculated shear-viscosity and thermalconductivity coefficients with molecular-dynamic results. Phys. Rev. A. 39 (12), 6391–6396 (1989).
dc.relation.references[62] Karkheck J., Stell G. Kinetic mean-field theories. J. Chem. Phys. 75 (3), 1475–1487 (1981).
dc.relation.references[63] Stell G., Karkheck J., van Beijeren H. Kinetic mean-field theories: Results of energy constraint in maximizing entropy. J. Chem. Phys. 79 (6), 3166–3167 (1983).
dc.relation.references[64] Karkheck J., Stell G. Maximization of entropy, kinetic equations, and irreversible thermodynamics. Phys. Rev. A. 25 (6), 3302–3327 (1982).
dc.relation.references[65] Karkheck J., Martina E., Stell G. Kinetic variational theory for mixtures: Kac-tail limit. Phys. Rev. A. 25 (6), 3328–3334 (1982).
dc.relation.references[66] Dyer K. M., Pettitt B. M., Stell G. Systematic investigation of theories of transport in the Lennard–Jones fluid. J. Chem. Phys. 126, 034502 (2007).
dc.relation.references[67] Morioka S. Phenomenological viscous factor in the nonequilibrium distribution function for liquids. Phys. Rev. E. 72 (5), 051203 (2005).
dc.relation.references[68] Karkheck J., Stell G., Xu J. Transport theory for the Lennard–Jones dense fluid. J. Chem. Phys. 89 (9), 5829–5833 (1988).
dc.relation.references[69] Zubarev D. N., Morozov V. G. Formulation of boundary conditions for the BBGKY hierarchy with allowance for local conservation laws. Theoretical and Mathematical Physics. 60, 814–820 (1984).
dc.relation.references[70] Zubarev D. N., Morozov V. G. Omelyan I. P., Tokarchuk M. V. Unification of kinetics and hydrodynamics in theory of transport phenomena. In: Collection of scientific works of Institute for Theoretical and Applied Mechanics of Siberian Branch of USSR Academy of Sciences. Models of mechanics of continuous media. Novosibirsk, 34–51 (1989), (in Russian).
dc.relation.references[71] Zubarev D. N., Morozov V. G., Omelyan I. P., Tokarchuk M. V. Kinetic equations for dense gases and liquids. Theoretical and Mathematical Physics. 87 (1), 113–129 (1991).
dc.relation.references[72] Zubarev D. N., Morozov V. G., Omelyan I. P., Tokarchuk M. V. Unification of the kinetic and hydrodynamic approaches in the theory of dense gases and liquids. Theoretical and Mathematical Physics. 96 (3), 997–1012 (1993).
dc.relation.references[73] Morozov V. G., Kobryn A. E., Tokarchuk M. V. Modified kinetic theory with consideration for slow hydrodynamical processes. Condens. Matter Phys. 4, 117–127 (1994).
dc.relation.references[74] Kobryn A. E., Morozov V. G., Omelyan I. P., Tokarchuk M. V. Enskog–Landau kinetic equation. Calculation of the transport coefficients for charged hard spheres. Physica A. 230 (1–2), 189–201 (1996).
dc.relation.references[75] Tokarchuk M. V., Omelyan I. P., Kobryn A. E. A consistent description of kinetics and hydrodynamics of systems of interacting particles by means of the nonequilibrium statistical operator method. Condens. Matter Phys. 1 (4), 687–751 (1998).
dc.relation.references[76] Kobryn A. E., Omelyan I. P., Tokarchuk M. V. The modified group expansions for constructions of solutions to the BBGKY hierarchy. J. Stat. Phys. 92 (5/6), 973–994 (1998).
dc.relation.references[77] Hlushak P. A., Tokarchuk M. V. A consistent description of kinetics and hydrodynamics of quantum Bosesystems. Condens. Matter Phys. 7 (3), 639–660 (2004).
dc.relation.references[78] Markiv B. B., Omelyan I. P., Tokarchuk M. V. On the problem of a consistent description of kinetic and hydrodynamic processes in dense gases and liquids. Condens. Matter Phys. 13 (2), 23005 (2010).
dc.relation.references[79] Markiv B. B., Omelyan I. P., Tokarchuk M. V. On the problem of a consistent description of kinetic and hydrodynamic processes in dense gases and liquids: Collective excitations spectrum. Condens. Matter Phys. 15 (1), 14001 (2012).
dc.relation.references[80] Zubarev D. N., Morozov V. G., Omelyan I. P., Tokarchuk M. V. The Enskog–Landau kinetic equation for charged hard spheres. In: Problems of atomic science and technique. Series: Nuclear physics investigations (theory and experiment). 3 (24), 60–65 (1992), (in Russian).
dc.relation.references[81] Green H. S. The Molecular Theory of Gases. Amsterdam, North Holland (1952).
dc.relation.references[82] Cohen E. G. D. Cluster expansion and hierarchy. Physica. 28 (10), 1045–1059 (1962).
dc.relation.references[83] Zubarev D. N., Novikov M. Yu. Generalized formulation of the boundary condition for the Liouville equation and for BBGKY hierarchy. Theoretical and Mathematical Physics. 13 (3), 406–420 (1972), (in Russian).
dc.relation.references[84] Cohen E. G. D. On the kinetic theory of dense gases. J. Math. Phys. 4 (2), 183–189 (1963).
dc.relation.references[85] Prigogine I. Non-Equilibrium Statistical Mechanics. J. Wiley and Sons, New York, Publie (1962)
dc.relation.references[86] Uhlenbeck G. E., Ford G. W. Lectures on Statistical Mechanics. (ed. M. Kac). American Mathematical Society, Providence RI (1963).
dc.relation.references[87] Gurov K. P. Fundamentals of kinetic theory. Moscow, Nauka (1966), (in Russian).
dc.relation.references[88] Vlasov A. A. Statistical distribution functions. Moscow, Nauka (1966), (in Russian).
dc.relation.references[89] Silin V. P. Introduction to the kinetic theory of gases. Moscow, Nauka (1971), (in Russian).
dc.relation.references[90] Zubarev D. N., Novikov M. Yu., R¨opke G. Die Boltzmangleichung and m¨ogliche Wege zur Entwicklung dynamischer Methoden in der Kinetishe Theorie. Fortschr. Physik. 21 (12), 703–734 (1973).
dc.relation.references[91] Zubarev D. N., Novikov M. Yu. Renormalized kinetic equations for a system with weak interaction and for a low-density gas. Theoretical and Mathematical Physics. 19 (2), 237–251 (1974), (in Russian).
dc.relation.references[92] Liboff R. L. Introduction to the Theory of Kinetic Equations. New York, John Willey and Sons (1969).
dc.relation.references[93] Balescu R. Equilibrium and nonequilibrium statistical mechanics. Research supported by EURATOM. New York, Wiley-Interscience (1975).
dc.relation.references[94] Balescu R. Statistical Mechanics of Charged Particles. Interscience Publishers (1963).
dc.relation.references[95] Ecker G. Theory of completely ionized plasma. Acad. Press New York and London (1972).
dc.relation.references[96] Yukhnovskii I. R., Holovko M. F. Statistical Theory of Classical Equilibrium Systems. Naukova Dumka, Kiev (1980), (in Russian).
dc.relation.references[97] Zubarev D. N. Statistical thermodynammics of turbulent transport processes. Theoretical and Mathematical Physics. 53 (1), 1004–1014 (1982).
dc.relation.references[98] Idzyk I. M., Ighatyuk V. V., Tokarchuk M. V. Fokker–Planck equation for nonequilibrium distribution function collective variables. I. Calculation of statistical weight, entropy, hydrodynamic speeds. Ukr. J. Phys. 41 (5–6), 596–604 (1996), (in Ukrainian).
dc.relation.references[99] Hlushak P., Tokarchuk M. Chain of kinetic equations for the distribution functions of particles in simple liquid taking into account nonlinear hydrodynamic fluctuations. Physica A. 443, 231–245 (2016).
dc.relation.references[100] Yukhnovskii I. R., Hlushak P. A., Tokarchuk M. V. BBGKY chain of kinetic equations, nonequilibrium statistical operator method and collective variable method in the statistical theory of nonequilibrium liquids. Condens. Matter Phys. 19, 43705 (2016).
dc.relation.references[101] Ramshaw J. D. Time-dependent direct correlashion function. Phys. Rev. A. 24 (3), 1567–1570 (1981).
dc.relation.references[102] Eu B. C. Dynamic Ornstein–Zernike Equation. In: Transport Coefficients of Fluids. Chemical physics. Vol. 82, 221–240 (2006).
dc.relation.references[103] Gan H. H., Eu B. C. Theory of the nonequilibrium structure of dense simple fluids: Effects of shearing. Phys. Rev. A. 45 (6), 3670–3686 (1992).
dc.relation.references[104] Brader J. M., Schmidt M. Nonequilibrium Ornstein–Zernike relation for Brownian many-body dynamics. J. Chem. Phys. 139 (10), 104108 (2013).
dc.relation.referencesen[1] Van der Waals J. D. On the Continuity of the Gas and Liquid State. Ph.D. Thesis, University of Leiden, Leiden (1873).
dc.relation.referencesen[2] Zwanzig R. W. High Temperature Equation of State by a Perturbation Method. I. Nonpolar Gases. J. Chem. Phys. 22 (8), 1420–1426 (1954).
dc.relation.referencesen[3] Rowlinson J. S., Widom B. Molecular Theory of Capillarity. Clarendon, Oxford, Chap. 1.6. (1982).
dc.relation.referencesen[4] Widom B. Intermolecular Forces and the Nature of the Liquid State. Science. 157 (3787), 375–382 (1967).
dc.relation.referencesen[5] Fisher M. E. Correlation Functions and the Critical Region of Simple Fluids. J. Math. Phys. 5, 944–962 (1964).
dc.relation.referencesen[6] Widom B. Equation of State in the Neighborhood of the Critical Point. J. Chem. Phys. 43, 3898–3905 (1965).
dc.relation.referencesen[7] Longuet-Higgens H. C., Widom B. A rigid sphere model for the melting of argon. Mol. Phys. 8 (6), 549–556 (1964).
dc.relation.referencesen[8] Barker J. A., Henderson D. Perturbation Theory and Equation of State for Fluids. II. A Successful Theory of Liquids. J. Chem. Phys. 47, 4714–4721 (1967).
dc.relation.referencesen[9] Weeks J. D., Chandler D., Andersen H. C. Role of Repulsive Forces in Determining the Equilibrium Structure of Simple Liquids. J. Chem. Phys. 54, 5237–5247 (1971).
dc.relation.referencesen[10] Kushickt J., Berne B. J. Role of attractive forces in self-diffusion in dense Lennard–Jones fluids. J. Chem. Phys. 59, 3732–3736 (1973).
dc.relation.referencesen[11] Chandler D. Equilibrium Structure and Molecular Motion in Liquids. Acc. Chem. Res. 7, 246–251 (1974).
dc.relation.referencesen[12] Barker J. A., Henderson D. What is "liquid"? Understanding the states of matter. Reviews of Modern Physics. 48 (4), 587–671 (1976).
dc.relation.referencesen[13] Dean D. P., Kushick J. N. On the role of attractive forces in the cage effect in simple liquids. J. Chem. Phys. 76 (1), 619–621 (1982).
dc.relation.referencesen[14] Sung W., Dahler J. S. Theory of transport processes in dense fluids. J. Chem Phys. 80 (7), 3025–3037 (1984).
dc.relation.referencesen[15] Barrat J. L., Hansen J. P. Basic Concepts for Simple and Complex Liquids. Cambridge University Press, Cambridge (2003).
dc.relation.referencesen[16] Hansen J. P., McDonald J. R. Theory of Simple Liquids. Academic, New York (2005).
dc.relation.referencesen[17] Chandler D. Interfaces and the driving force of hydrophobic assembly. Nature. 437, 640–646 (2005).
dc.relation.referencesen[18] Kirchner B. Theory of Complicated Liquids: Investigation of Liquids, Solvents and Solvent Effects with Modern Theoretical Methods. Phys. Rep. 440 (1–3), 1–111 (2007).
dc.relation.referencesen[19] Berthier L., Tarjus G. Nonperturbative Effect of Attractive Forces in Viscous Liquids. Phys. Rev. Lett. 103, 170601 (2009).
dc.relation.referencesen[20] Melnyk R., Nezbeda I., Henderson D., Trokhymchuk A. On the role of the reference system in perturbation theory: An augmented van der Waals theory of simple fluids. Fluid Phase Equilibria. 279 (1), 1–10 (2009).
dc.relation.referencesen[21] Pedersen U. R., Schroder T. B., Dyre J. C. Repulsive Reference Potential Reproducing the Dynamics of a Liquid with Attractions. Phys. Rev. Lett. 105 (15), 157801 (2010).
dc.relation.referencesen[22] Toxvaerd S., Dyre J. C. Communication: Shifted forces in molecular dynamics. J. Chem. Phys. 134 (8), 081102 (2011).
dc.relation.referencesen[23] Toxvaerd S., Dyre J. C. Role of the first coordination shell in determining the equilibrium structure and dynamics of simple liquids. J. Chem. Phys. 135 (13), 134501 (2011).
dc.relation.referencesen[24] Berthier L., Tarjus G. The role of attractive forces in viscous liquids. J. Chem. Phys. 134 (21), 214503 (2011).
dc.relation.referencesen[25] Ingebrigtsen T. S., Schrøder T. B., Dyre J. C. What Is a Simple Liquid? Phys. Rev. X. 2 (1), 011011 (2012).
dc.relation.referencesen[26] Hordiichuk V., Trokhymchuk A., Skvara J., Nezbed I. Excluded volume interaction in a Lennard–Jones fluid. Preprint of the Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, ICMP-18-07E (2018).
dc.relation.referencesen[27] Mryglod I. M., Omelyan I. P., Tokarchuk M. V. Generalized Collective Modes for the Lennard–Jones fluid. Mol. Phys. 84 (2), 235–259 (1995).
dc.relation.referencesen[28] Mryglod I. M., Folk R., Tokarchuk M. V. On the hydrodynamic theory of a magnetic liquid. I. General description. Physica A. 220 (3–4), 325–348 (1995).
dc.relation.referencesen[29] Mryglod I. M., Omelyan I. P., Tokarchuk M. V. Wavevector- and frequency-dependent shear viscosity of water: the modified collective mode approach and molecular dynamics calculations. Condens. Matter Phys. 8 (1), 25–46 (2005).
dc.relation.referencesen[30] Markiv B., Omelyan I., Tokarchuk M. Consistent description of kinetics and hydrodynamics of weakly nonequilibrium processes in simple liquids. J. Stat. Phys. 155, 843–866 (2014).
dc.relation.referencesen[31] Bogolubov N. N. Problems of a dynamical theory in statistical physics. In: Studies in statistical mechanics. Vol. 1. Amsterdam, North-Holland (1962).
dc.relation.referencesen[32] Chapman S., Cowling T. The mathematical theory of non-uniform gases: an account of the kinetic theory of viscosity, thermal conduction and diffusion in gases. Cambridge, University Press, 3rd ed. (1970).
dc.relation.referencesen[33] Ferziger J. H., Kaper H. G. Mathematical Theory of Transport Processes in Gases. North-Holland, Amsterdam (1972).
dc.relation.referencesen[34] Resibois P., de Leener M. Classical Kinetic Theory of Fluids. New York, John Willey and Sons (1977).
dc.relation.referencesen[35] Humenyuk Y. A., Tokarchuk M. V. Extension of Hydrodynamic Balance Equations for Simple Fluids. J. Stat. Phys. 142, 1052–1084 (2011).
dc.relation.referencesen[36] Enskog D. Kinetische Theorie der Warmeleitung Reibung und Selbstdiffusion in gevissen verdichtetten Gasen und Flussigkeiten. Svenska Vetenskapsaked. Handl. 63, 4 (1922).
dc.relation.referencesen[37] Van Beijeren H., Ernst M. H. The modified Enskog equation. Physica. 68 (3), 437–456 (1973).
dc.relation.referencesen[38] Van Beijeren H., Ernst M. H. The modified Enskog equation for mixtures. Physica. 70 (2), 225–242 (1973).
dc.relation.referencesen[39] Polewczak J., Stell G. Transport Coefficients in Some Stochastic Models of the Revised Enskog Equation. J. Stat. Phys. 109, 569–590 (2002).
dc.relation.referencesen[40] Polewczak J. A review of the kinetic modelings for non-reactive and reactive dense fluids: Questions and problems. Rivista di Matematica della Universita’ di Parma. 4, 23–55 (2001).
dc.relation.referencesen[41] Polewczak J. Hard-sphere kinetic models for inert and reactive mixtures. J. Phys., Condens. Matter. 28, 414022 (2016).
dc.relation.referencesen[42] Davis H. T., Rice S. A., Sengers J. V. On the kinetic theory of dense fluids. IX. The fluid of rigid spheres with a square-well attraction. J. Chem. Phys. 35 (6), 2210–2233 (1961).
dc.relation.referencesen[43] Karkheck J., van Beijeren H., de Schepper I., Stell G. Kinetic theory and H theorem for a dense squarewell fluid. Phys. Rev. A. 32 (4), 2517–2520 (1985).
dc.relation.referencesen[44] Wilbertz H., Micheis J., van Beijeren H., Leegwater J. A. Self-Diffusion of Particles Interacting Through a Square-Well or Square-Shoulder Potential. J. Stat. Phys. 53 (5/6), 1155–1177 (1988).
dc.relation.referencesen[45] Van Beijeren H., Karkheck J., Sengers J. V. Nonequilibrium temperature and bulk viscosity for a dense fluid of square-well molecules. Phys. Rev. A. 37 (6), 2247–2250 (1988).
dc.relation.referencesen[46] Leegwater J. A., van Beijeren H., Michels P. J. Linear kinetic theory of the square-well fliud. J. Phys., Condens. Matter. 1 (1), 237–255 (1989).
dc.relation.referencesen[47] Dufty J. W., Mo K. C., Gubbins K. E. Models for self-diffusion in the square well fluid. J. Chem. Phys. 94 (4), 3132–3140 (1991).
dc.relation.referencesen[48] Garland G. E., Dufty J. W. Bound state contributions to transport coefficients. J. Chem. Phys. 95 (4), 2702–2716 (1991).
dc.relation.referencesen[49] Yu Y.-X., Han M.-H., Gao G.-H. Self-di†usion in a Nuid of square-well spheres. Phys. Chem. Chem. Phys. 3, 437–443 (2001).
dc.relation.referencesen[50] Dwivedee D. K., Srivastava R., Khanna K. N. Transport coefficients of square-well fluids. Fluid Phase Equilibria. 263 (2), 199–204 (2008).
dc.relation.referencesen[51] Takada S., Saitoh K., Hayakawa H. Kinetic theory for dilute cohesive granular gases with a square-well potential. Phys. Rev. E. 94 (1), 012906 (2016).
dc.relation.referencesen[52] Tokarchuk M. V., Omelyan I. P. To the kinetic theory of processes transfer in dense gases. Preprint of the Institute for Theoretical Physics of the Academy of Sciences of Ukraine, ITR-87-152R (1987), (in Russian).
dc.relation.referencesen[53] Tokarchuk M. V., Omelyan I. P. The kinetic equation for the systems with multistage interparticle potential. H-theorem. Preprint of the Institute for Theoretical Physics of the Academy of Sciences of Ukraine, ITR-89-49R (1989).
dc.relation.referencesen[54] Tokarchuk M. V., Omelyan I. P. Model kinetic equations for dense gases and liquids. Ukr. Phys. J. 35 (8), 1255–1261 (1990).
dc.relation.referencesen[55] Omelyan I. P., Tokarchuk M. V. Kinetic equation for liquids with a multistep potential of interaction. H-theorem. Physica A. 234 (1–2), 89–107 (1996).
dc.relation.referencesen[56] Tokarchuk M. V., Omelyan I. P., Kobryn O. E. Kinetic equation for liquids ith a multistep potential of interaction: Calculation of transport coefficient. Phys. Rev. E. 62 (6), 8021–8036 (2000).
dc.relation.referencesen[57] Tokarchuk M. V., Humenyuk Y. A. Hydrodynamic equations for dense fluid mixtures with multistep interaction between particles. Condens. Matter Phys. 10 (2), 151–163 (2007).
dc.relation.referencesen[58] Tokarchuk M. V., Humenyuk Y. A. Derivation of hydrodynamic equations for dense gaseous mixtures with multi-step interaction between particles. Preprint of the Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, ISMR-04-09U (2004), (in Ukrainian).
dc.relation.referencesen[59] Humenyuk Y. A., Tokarchuk M. V. Limit behavior of kinetic theory for a system with multistage potential. Ukr. Phys. J. 55 (4), 453–460 (1999), (in Ukrainian).
dc.relation.referencesen[60] Rice S. A., Allnatt A. R. On the kinetic theory of dense fluids. VI. Singlet distribution function for rigid spheres with an attractive potential. J. Chem. Phys. 34 (6), 2144–2155 (1961).
dc.relation.referencesen[61] Vogelsang R., Hoheisel C. Rice-Allnatt theory: Comparison of calculated shear-viscosity and thermalconductivity coefficients with molecular-dynamic results. Phys. Rev. A. 39 (12), 6391–6396 (1989).
dc.relation.referencesen[62] Karkheck J., Stell G. Kinetic mean-field theories. J. Chem. Phys. 75 (3), 1475–1487 (1981).
dc.relation.referencesen[63] Stell G., Karkheck J., van Beijeren H. Kinetic mean-field theories: Results of energy constraint in maximizing entropy. J. Chem. Phys. 79 (6), 3166–3167 (1983).
dc.relation.referencesen[64] Karkheck J., Stell G. Maximization of entropy, kinetic equations, and irreversible thermodynamics. Phys. Rev. A. 25 (6), 3302–3327 (1982).
dc.relation.referencesen[65] Karkheck J., Martina E., Stell G. Kinetic variational theory for mixtures: Kac-tail limit. Phys. Rev. A. 25 (6), 3328–3334 (1982).
dc.relation.referencesen[66] Dyer K. M., Pettitt B. M., Stell G. Systematic investigation of theories of transport in the Lennard–Jones fluid. J. Chem. Phys. 126, 034502 (2007).
dc.relation.referencesen[67] Morioka S. Phenomenological viscous factor in the nonequilibrium distribution function for liquids. Phys. Rev. E. 72 (5), 051203 (2005).
dc.relation.referencesen[68] Karkheck J., Stell G., Xu J. Transport theory for the Lennard–Jones dense fluid. J. Chem. Phys. 89 (9), 5829–5833 (1988).
dc.relation.referencesen[69] Zubarev D. N., Morozov V. G. Formulation of boundary conditions for the BBGKY hierarchy with allowance for local conservation laws. Theoretical and Mathematical Physics. 60, 814–820 (1984).
dc.relation.referencesen[70] Zubarev D. N., Morozov V. G. Omelyan I. P., Tokarchuk M. V. Unification of kinetics and hydrodynamics in theory of transport phenomena. In: Collection of scientific works of Institute for Theoretical and Applied Mechanics of Siberian Branch of USSR Academy of Sciences. Models of mechanics of continuous media. Novosibirsk, 34–51 (1989), (in Russian).
dc.relation.referencesen[71] Zubarev D. N., Morozov V. G., Omelyan I. P., Tokarchuk M. V. Kinetic equations for dense gases and liquids. Theoretical and Mathematical Physics. 87 (1), 113–129 (1991).
dc.relation.referencesen[72] Zubarev D. N., Morozov V. G., Omelyan I. P., Tokarchuk M. V. Unification of the kinetic and hydrodynamic approaches in the theory of dense gases and liquids. Theoretical and Mathematical Physics. 96 (3), 997–1012 (1993).
dc.relation.referencesen[73] Morozov V. G., Kobryn A. E., Tokarchuk M. V. Modified kinetic theory with consideration for slow hydrodynamical processes. Condens. Matter Phys. 4, 117–127 (1994).
dc.relation.referencesen[74] Kobryn A. E., Morozov V. G., Omelyan I. P., Tokarchuk M. V. Enskog–Landau kinetic equation. Calculation of the transport coefficients for charged hard spheres. Physica A. 230 (1–2), 189–201 (1996).
dc.relation.referencesen[75] Tokarchuk M. V., Omelyan I. P., Kobryn A. E. A consistent description of kinetics and hydrodynamics of systems of interacting particles by means of the nonequilibrium statistical operator method. Condens. Matter Phys. 1 (4), 687–751 (1998).
dc.relation.referencesen[76] Kobryn A. E., Omelyan I. P., Tokarchuk M. V. The modified group expansions for constructions of solutions to the BBGKY hierarchy. J. Stat. Phys. 92 (5/6), 973–994 (1998).
dc.relation.referencesen[77] Hlushak P. A., Tokarchuk M. V. A consistent description of kinetics and hydrodynamics of quantum Bosesystems. Condens. Matter Phys. 7 (3), 639–660 (2004).
dc.relation.referencesen[78] Markiv B. B., Omelyan I. P., Tokarchuk M. V. On the problem of a consistent description of kinetic and hydrodynamic processes in dense gases and liquids. Condens. Matter Phys. 13 (2), 23005 (2010).
dc.relation.referencesen[79] Markiv B. B., Omelyan I. P., Tokarchuk M. V. On the problem of a consistent description of kinetic and hydrodynamic processes in dense gases and liquids: Collective excitations spectrum. Condens. Matter Phys. 15 (1), 14001 (2012).
dc.relation.referencesen[80] Zubarev D. N., Morozov V. G., Omelyan I. P., Tokarchuk M. V. The Enskog–Landau kinetic equation for charged hard spheres. In: Problems of atomic science and technique. Series: Nuclear physics investigations (theory and experiment). 3 (24), 60–65 (1992), (in Russian).
dc.relation.referencesen[81] Green H. S. The Molecular Theory of Gases. Amsterdam, North Holland (1952).
dc.relation.referencesen[82] Cohen E. G. D. Cluster expansion and hierarchy. Physica. 28 (10), 1045–1059 (1962).
dc.relation.referencesen[83] Zubarev D. N., Novikov M. Yu. Generalized formulation of the boundary condition for the Liouville equation and for BBGKY hierarchy. Theoretical and Mathematical Physics. 13 (3), 406–420 (1972), (in Russian).
dc.relation.referencesen[84] Cohen E. G. D. On the kinetic theory of dense gases. J. Math. Phys. 4 (2), 183–189 (1963).
dc.relation.referencesen[85] Prigogine I. Non-Equilibrium Statistical Mechanics. J. Wiley and Sons, New York, Publie (1962)
dc.relation.referencesen[86] Uhlenbeck G. E., Ford G. W. Lectures on Statistical Mechanics. (ed. M. Kac). American Mathematical Society, Providence RI (1963).
dc.relation.referencesen[87] Gurov K. P. Fundamentals of kinetic theory. Moscow, Nauka (1966), (in Russian).
dc.relation.referencesen[88] Vlasov A. A. Statistical distribution functions. Moscow, Nauka (1966), (in Russian).
dc.relation.referencesen[89] Silin V. P. Introduction to the kinetic theory of gases. Moscow, Nauka (1971), (in Russian).
dc.relation.referencesen[90] Zubarev D. N., Novikov M. Yu., R¨opke G. Die Boltzmangleichung and m¨ogliche Wege zur Entwicklung dynamischer Methoden in der Kinetishe Theorie. Fortschr. Physik. 21 (12), 703–734 (1973).
dc.relation.referencesen[91] Zubarev D. N., Novikov M. Yu. Renormalized kinetic equations for a system with weak interaction and for a low-density gas. Theoretical and Mathematical Physics. 19 (2), 237–251 (1974), (in Russian).
dc.relation.referencesen[92] Liboff R. L. Introduction to the Theory of Kinetic Equations. New York, John Willey and Sons (1969).
dc.relation.referencesen[93] Balescu R. Equilibrium and nonequilibrium statistical mechanics. Research supported by EURATOM. New York, Wiley-Interscience (1975).
dc.relation.referencesen[94] Balescu R. Statistical Mechanics of Charged Particles. Interscience Publishers (1963).
dc.relation.referencesen[95] Ecker G. Theory of completely ionized plasma. Acad. Press New York and London (1972).
dc.relation.referencesen[96] Yukhnovskii I. R., Holovko M. F. Statistical Theory of Classical Equilibrium Systems. Naukova Dumka, Kiev (1980), (in Russian).
dc.relation.referencesen[97] Zubarev D. N. Statistical thermodynammics of turbulent transport processes. Theoretical and Mathematical Physics. 53 (1), 1004–1014 (1982).
dc.relation.referencesen[98] Idzyk I. M., Ighatyuk V. V., Tokarchuk M. V. Fokker–Planck equation for nonequilibrium distribution function collective variables. I. Calculation of statistical weight, entropy, hydrodynamic speeds. Ukr. J. Phys. 41 (5–6), 596–604 (1996), (in Ukrainian).
dc.relation.referencesen[99] Hlushak P., Tokarchuk M. Chain of kinetic equations for the distribution functions of particles in simple liquid taking into account nonlinear hydrodynamic fluctuations. Physica A. 443, 231–245 (2016).
dc.relation.referencesen[100] Yukhnovskii I. R., Hlushak P. A., Tokarchuk M. V. BBGKY chain of kinetic equations, nonequilibrium statistical operator method and collective variable method in the statistical theory of nonequilibrium liquids. Condens. Matter Phys. 19, 43705 (2016).
dc.relation.referencesen[101] Ramshaw J. D. Time-dependent direct correlashion function. Phys. Rev. A. 24 (3), 1567–1570 (1981).
dc.relation.referencesen[102] Eu B. C. Dynamic Ornstein–Zernike Equation. In: Transport Coefficients of Fluids. Chemical physics. Vol. 82, 221–240 (2006).
dc.relation.referencesen[103] Gan H. H., Eu B. C. Theory of the nonequilibrium structure of dense simple fluids: Effects of shearing. Phys. Rev. A. 45 (6), 3670–3686 (1992).
dc.relation.referencesen[104] Brader J. M., Schmidt M. Nonequilibrium Ornstein–Zernike relation for Brownian many-body dynamics. J. Chem. Phys. 139 (10), 104108 (2013).
dc.rights.holder© Національний університет “Львівська політехніка”, 2022
dc.subjectкінетичні рівняння
dc.subjectнерівноважний статистичний оператор
dc.subjectфункція розподілу
dc.subjectпроста рідина
dc.subjectkinetic equations
dc.subjectnon-equilibrium statistical operator
dc.subjectdistribution function
dc.subjectsimple fluid
dc.titleTo the kinetic theory of dense gases and liquids. Calculation of quasi-equilibrium particle distribution functions by the method of collective variables
dc.title.alternativeДо кінетичної теорії густих газів та рідин. Розрахунок квазірівноважних функцій розподілу частинок методом колективних змінних
dc.typeArticle

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Mathematical Modeling And Computing. – 2022. – Vol. 9, No. 2