The Method of Collective Variables in the Theory of Nonlinear Fluctuations with Account for Kinetic Processes
DOI:
https://doi.org/10.15407/ujpe67.8.579Keywords:
simple fluid, nonlinear fluctuations, non-equilibrium statistical operator, distribution function, Fokker–Planck equationAbstract
The set of parameters of the Bogolyubov reduced description, which includes collective variables, has been optimized for the consistent description of the kinetics and hydrodynamics of the systems of interacting particles. The contributions from short- and long-range interactions between the particles are separated. The short-range interactions (for example, the hard-sphere model) are described in the coordinate-momentum space, and the long-range ones in the space of collective variables. The short-range component is considered to be basic. Using the method of Zubarev non-equilibrium statistical operator, a system of transport equations for the non-equilibrium one-particle distribution function, the non-equilibrium average value for the density of particle interaction energy, and the non-equilibrium distribution function of collective variables are obtained. The applied method of collective variables allowed both the structural function and the hydrodynamic velocities of collective variables to be calculated in approximations higher than the Gaussian one.
References
P. R'esibois, M. de Leener, Classical Kinetic Theory of fluids (John Wiley and Sons, 1977).
J. Boon, S. Yip, Molecular Hydrodynamics (McGraw-Hill Inc., 1980).
G. R¨opke, Nonequilibrium Statistical Physics (Wiley-VCH Verlag GmbH and Co., 2013).
Yu.L. Klimontovich. Turbulent Motion and the Structure of Chaos. A New Approach to the Statistical Theory of Open Systems (Kluwer Academic, 1991).
https://doi.org/10.1007/978-94-011-3426-2_7
U. Balucani, M. Zoppi. Dynamics of the Liquid State (Clarendon Press, 1994).
R. Balescu. Statistical Dynamics: Matter out of Equilibrium (World Scientific, 1997).
R. Zwanzig. Nonequilibrium Statistical Mechanics (Oxford University Press, 2001).
D. Zubarev, V. Morozov, G. R¨opke. Statistical Mechanics of Nonequilibrium Processes (Akademie, 1996), Vol. 1.
D. Zubarev, V. Morozov, G. R¨opke. Statistical Mechanics of Nonequilibrium Processes (Akademie, 1996), Vol. 2.
G. Mazenko. Nonequilibrium Statistical Mechanics (WileyVCH Verlag GmbH. and Co., 2006).
https://doi.org/10.1002/9783527618958
Yu.L. Klimontovich, H. Wilhelmsson, A.G. Zagorodnii, I.P. Yakimenko. Statistical Theory of Confined PlasmaMolecular Systems (Moscow State University, 1990) (in Russian).
Y.L. Klimontovich, D. Kremp, W.D Kraeft. Advances in Chemistry and Physics (Wiley, 2007).
L.A. Bulavin. Neutron Diagnostics of Liquid Matter State (Institute for Safety Problems of Nuclear Power Plants, 2012) (in Ukrainian).
M. Bonitz, J. Lopez, K. Becker, H. Thomsen., Complex Plasmas. Scientific Challenges and Technological Opportunities (Springer, 2014).
https://doi.org/10.1007/978-3-319-05437-7
S.V. Peletminskii, Yu.V. Slyusarenko, A.I. Sokolovsky. Kinetics and hydrodynamics of long-wave fluctuations under external random force. Physica A 326, 412 (2003).
https://doi.org/10.1016/S0378-4371(03)00255-3
S.O. Nikolayenko, Yu.V. Slyusarenko. Microscopic theory of relaxation processes in systems of particles interacting with the hydrodynamic medium. J. Math. Phys. 50, 083305 (2009).
https://doi.org/10.1063/1.3204080
O.Yu. Slyusarenko, A.V. Chechkin, Yu.V. Slyusarenko. The Bogolyubov-Born-Green-Kirkwood-Yvon hierarchy and Fokker-Planck equation for many-body dissipative randomly driven systems. J. Math. Phys. 56, 043302 (2015).
https://doi.org/10.1063/1.4918612
Y.A. Humenyuk, M.V. Tokarchuk. Extension of hydrodynamic balance equations for simple fluids. J. Stat. Phys. 142, 1052 (2011).
https://doi.org/10.1007/s10955-011-0141-y
K. Yoshida, T. Arimitsu. Inertial-range structure of Gross-Pitaevskii turbulence within a spectral closure approximation. J. Phys. A: Math. Theor. 46, 335501 (2013).
https://doi.org/10.1088/1751-8113/46/33/335501
P. Mendoza-Mendez, L. Lopez-Flores, A. Vizcarra-Redon, L.F. Sanchez-Diaz, M. Medina-Noyola. Generalized Langevin equation for tracer diffusion in atomic liquids. Physica A 394, 1 (2014).
https://doi.org/10.1016/j.physa.2013.09.061
J.P. Boon, J.F. Lutsko, C. Lutsko. Microscopic approach to nonlinear reaction-diffusion: The case of morphogen gradient formation. Phys. Rev. E 85, 021126 (2012).
https://doi.org/10.1103/PhysRevE.85.021126
G.F. Mazenko. Fundamental theory of statistical particle dynamics. Phys. Rev. E 81, 061102 (2010).
https://doi.org/10.1103/PhysRevE.81.061102
G.F. Mazenko. Smoluchowski dynamics and the ergodicnonergodic transition. Phys. Rev. E 83, 041125 (2011).
https://doi.org/10.1103/PhysRevE.83.041125
P. Kostrobij, R. Tokarchuk, M. Tokarchuk, V. Markiv. Zubarev nonequilibrium statistical operator method in Renyi statistics. Reaction-diffusion processes. Condens. Matter Phys. 17, 33005 (2014).
https://doi.org/10.5488/CMP.17.33005
P.A. Hlushak, M.V. Tokarchuk. Quantum transport equations for Bose systems taking into account nonlinear hydrodynamic processes. Condens. Matter Phys. 17, 23606 (2014).
https://doi.org/10.5488/CMP.17.23606
C.A.B. Silva, A.R. Vasconcellos, J.G. Ramos, R. Luzzi. Generalized kinetic equation for far-from-equilibrium many-body systems. J. Stat. Phys. 143, 1020 (2011).
https://doi.org/10.1007/s10955-011-0222-y
V.N. Tsytovich, U. de Andelis. Kinetic theory of dusty plasmas. V. The hydrodynamic equations. Phys. Plasmas 11, 496 (2004).
https://doi.org/10.1063/1.1634255
A.I. Olemskoi. Theory of structure transformations in nonequilibrium condensed matter. Horizons in World Physics Series. Vol. 231. (NOVA Science Publishers, 1999).
B. Markiv, R. Tokarchuk, P. Kostrobij, M. Tokarchuk. Nonequilibrium statistical operator method in Renyi statistics. Physica A 390, 785 (2011).
https://doi.org/10.1016/j.physa.2010.11.009
I.M. Mryglod, M.V. Tokarchuk. On statistical hydrodynamics of simple liquds. In: Problems of Atomic Science and Technique. Series: Nuclear Physics Investigations (Theory and Experiment) (Kharkov Physico-Technical Institute, 1992), 3 (24), p. 134.
I.M. Mryglod, I.P. Omelyan, M.V. Tokarchuk. Generalized collective modes for the Lennard-Jones fluid. Mol. Phys. 84, 235 (1995).
https://doi.org/10.1080/00268979500100181
B.B. Markiv, I.P. Omelyan, M.V. Tokarchuk. Relaxation to the state of molecular hydrodynamics in the generalized hydrodynamics of liquids. Phys. Rev. E 82, 041202 (2010).
https://doi.org/10.1103/PhysRevE.82.041202
D.N. Zubarev, V.G. Morozov, I.P. Omelyan, M.V. Tokarchuk. Unification of the kinetic and hydrodynamic approaches in the theory of dense gases and liquids. Theor. Math. Phys. 96, 997 (1993).
https://doi.org/10.1007/BF01019063
M.V. Tokarchuk, I.P. Omelyan, A.E. Kobryn. A consistent description of kinetics and hydrodynamics of systems of interacting particles by means of the nonequilibrium statistical operator method. Condens. Matter Phys. 1, 687 (1998).
https://doi.org/10.5488/CMP.1.4.687
A.E. Kobryn, I.P. Omelyan, M.V. Tokarchuk. The modified group expansions for construction of solutions to the BBGKY hierarchy. J. Stat. Phys. 92, 973 (1998).
https://doi.org/10.1023/A:1023044610690
B. Markiv, I. Omelyan, M. Tokarchuk. Consistent description of kinetics and hydrodynamics of weakly nonequilibrium processes in simple liquids. J. Stat. Phys. 155, 843 (2014).
https://doi.org/10.1007/s10955-014-0980-4
B. Markiv, M. Tokarchuk. Consistent description of kinetics and hydrodynamics of dusty plasma. Phys. Plasmas 21, 023707 (2014).
https://doi.org/10.1063/1.4865581
J.R. Dorfman. Advances and challenges in the kinetic theory of gases. Physica A 106, 77 (1981).
https://doi.org/10.1016/0378-4371(81)90208-9
Yu.L. Klimontovich. On the need for and the possibility of a unified description of kinetic and hydrodynamic processes. Theor. Math. Phys. 92, 909 (1992).
https://doi.org/10.1007/BF01015557
Yu.L. Klimontovich. The unified description of kinetic and hydrodynamic processes in gases and plasmas. Phys. Lett. A 170, 434 (1992).
https://doi.org/10.1016/0375-9601(92)90747-A
E.G.D. Cohen. Fifty years of kinetic theory. Physica A 194, 229 (1993).
https://doi.org/10.1016/0378-4371(93)90357-A
S.K. Schnyder, F. Hofling, T. Franosch, Th. Voigtmann. Long-wavelength anomalies in the asymptotic behavior of mode-coupling theory. J Phys.: Condens. Matter. 23, 234121 (2011).
https://doi.org/10.1088/0953-8984/23/23/234121
T. Franosch. Long-time limit of correlation functions. J. Phys. A: Math. Theor. 47, 325004 (2014).
https://doi.org/10.1088/1751-8113/47/32/325004
D.N. Zubarev, V.G. Morozov. Formulation of boundary conditions for the BBGKY hierarchy with allowance for local conservation laws. Theor. Math. Phys. 60, 814 (1984).
https://doi.org/10.1007/BF01018982
D.N. Zubarev, V.G. Morozov, I.P. Omelyan, M.V. Tokarchuk. Kinetic equations for dense gases and liquids. Theor. and Math. Phys. 87, 412 (1991).
https://doi.org/10.1007/BF01016582
A.E. Kobryn, V.G. Morozov, I.P. Omelyan, M.V. Tokarchuk. Enskog-Landau kinetic equation. Calculation of the transport coefficients for charged hard spheres. Physica A 230, 189 (1996).
https://doi.org/10.1016/0378-4371(96)00044-1
D.N. Zubarev, V.G. Morozov. Nonequilibrium statistical ensembles in kinetic theory and hydrodynamics. In: Collection of Scientific Works of Mathematical Institute of USSR Academy of Sciences (Nauka, 1989), 191, p. 140 (in Russian).
V.G. Morozov, A.E. Kobryn, M.V. Tokarchuk. Modified kinetic theory with consideration for slow hydrodynamical processes. Condens. Matter Phys. 4, 117 (1994).
https://doi.org/10.5488/CMP.4.117
P. Hlushak, M. Tokarchuk. Chain of kinetic equations for the distribution functions of particles in simple liquid taking into account nonlinear hydrodynamic fluctuations. Physica A 443, 231 (2016).
https://doi.org/10.1016/j.physa.2015.09.059
I.R. Yukhnovskii, P.A. Hlushak, M.V. Tokarchuk. BBGKY chain of kinetic equations, non-equilibrium statistical operator method and collective variable method in the statistical theory of non-equilibrium liquids. Condens. Matter Phys. 19, 43705 (2016).
https://doi.org/10.5488/CMP.19.43705
I.R. Yukhnovskii, M.F. Holovko. Statistical Theory of Classical Equilibrium Systems ( Naukova Dumka, 1980) (in Russian).
D.N. Zubarev, A.M. Khazanov. Generalized Fokker-Planck equation and construction of projection operators for different methods of reduced description of nonequilibrium states. Theor. Math. Phys. 34, 43 (1978).
https://doi.org/10.1007/BF01036470
K. Kawasaki. In Phase Transition and Critical Phenomena. Edited by C. Domb, M.S. Green (Acad. Press, 1976), Vol. 5A, p. 165.
D.N. Zubarev. Statistical thermodynammics of turbulent transport processes. Theor. Math. Phys. 53, 1004 (1982).
https://doi.org/10.1007/BF01014797
I.M. Idzyk, V.V. Ighatyuk, M.V. Tokarchuk. Fokker-Planck equation for nonequilibrium distribution function of collective variables. I. Calculation of the statistical weight, entropy, hydrodynamic velocities. Ukr. J. Phys. 41, 596 (1996).
M.V. Ignatiuk, V.V. Tokarchuk. Statistical theory of nonlinear hydrodynamic fluctuations in ionic systems. Theor. Math. Phys. 108, 1208 (1996).
https://doi.org/10.1007/BF02070247
V.G. Morozov. Low-frequency correlation functions in case of nonlinear dynamics of fluctuations. Physica A 110, 201 (1982).
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