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

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.