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 TMF, 2003, Volume 134, Number 3, Pages 487–500 (Mi tmf166)

Thermodynamics and Hydrodynamics (Statistical Foundations): 3. Hydrodynamic Equations

G. A. Martynov

Institute of Physical Chemistry, Russian Academy of Sciences

Abstract: We show that all the hydrodynamic equations can be obtained from the BBGKY hierarchy. The theory is constructed by expanding the distribution functions in series in a small parameter $\varepsilon=R/L\leq10^{-8}$, where$R\approx10^{-7}$cm is the radius of the correlation sphere and $L$ is the characteristic macroscopic dimension. We also show that in the zeroth-order approximation with respect to this parameter, the BBGKY hierarchy implies the local equilibrium and the transport equations for the ideal Euler fluid; in the first-order approximation with respect to $\varepsilon$, the BBGKY hierarchy implies the hydrodynamic equations for viscous fluids. Moreover, we prove that the intrinsic energy flux must include both the kinetic energy flux proportional to the temperature gradient and the potential energy flux proportional to the density gradient. We show that the hydrodynamic equations hold for $t\gg\tau\approx10^{-12}$s and $L\gg R\approx10^{-7}$cm.

Keywords: BBGKY hierarchy, conservation laws, small parameter, hydrodynamic equations

DOI: https://doi.org/10.4213/tmf166

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English version:
Theoretical and Mathematical Physics, 2003, 134:3, 427–438

Bibliographic databases:

Revised: 19.02.2002

Citation: G. A. Martynov, “Thermodynamics and Hydrodynamics (Statistical Foundations): 3. Hydrodynamic Equations”, TMF, 134:3 (2003), 487–500; Theoret. and Math. Phys., 134:3 (2003), 427–438

Citation in format AMSBIB
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\by G.~A.~Martynov
\paper Thermodynamics and Hydrodynamics (Statistical Foundations): 3.~Hydrodynamic Equations
\jour TMF
\yr 2003
\vol 134
\issue 3
\pages 487--500
\mathnet{http://mi.mathnet.ru/tmf166}
\crossref{https://doi.org/10.4213/tmf166}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2001820}
\zmath{https://zbmath.org/?q=an:1178.82067}
\transl
\jour Theoret. and Math. Phys.
\yr 2003
\vol 134
\issue 3
\pages 427--438
\crossref{https://doi.org/10.1023/A:1022609724144}

• http://mi.mathnet.ru/eng/tmf166
• https://doi.org/10.4213/tmf166
• http://mi.mathnet.ru/eng/tmf/v134/i3/p487

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This publication is cited in the following articles:
1. G. A. Martynov, “Thermodynamics and Hydrodynamics (Statistical Foundations): 4. Kinetic Processes in Multicomponent Systems”, Theoret. and Math. Phys., 136:2 (2003), 1167–1176
2. G. A. Martynov, “Statistical Theory of Equilibrium Fluctuations”, Theoret. and Math. Phys., 139:2 (2004), 706–725
3. G. A. Martynov, “General Theory of Acoustic Wave Propagation in Liquids and Gases”, Theoret. and Math. Phys., 146:2 (2006), 285–294
4. Martynov, GA, “The Ornstein-Zernike equation and critical phenomena in fluids”, Journal of Chemical Physics, 129:24 (2008), 244509
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