Microscopic classic hydrodynamics and methods of averaging
Leonid Stefanovich Kuz'menkov, Pavel A. Andreev · 2012
In classic mechanics there are at least two general methods for hydrodynamic equation derivation. The flrst one is the consideration of macroscopic parallelogram in a medium and calculation for number of particles, momentum or energy passing through the planes. It is the way of macroscopic hydrodynamics. The second method of the hydrodynamic equations derivation includes the physical kinetics and ofiers the microscopic method starting from Liouville equation. Using of the physical kinetics as intermediate step leads to additional unnecessary limitation. This method includes derivation of the kinetic equation chain which also called BBGKY chain. The hydrodynamic equation arises as moments of the frequency function. We present a new method for derivation of the hydrodynamic equation directly from classic motion equations. The flrst and most important step is the deflnition of the particles density in vicinity of the point of the physical space. This deflnition contains generalized function (Dirac's delta function) and represents analytical representation of a notion of the physically inflnitesimal volume. The Newton law of motion is used for motion description of each particle in vicinity of point of physical space. Particle density is the flrst collective variable in our description. Next step is the derivation of equation of temporal evolution for the particle density. In the result new physical quantities are appeared and we can derive evolution equations for these quantities. During derivation a chain of equations is arisen, this chain is an analog of the BBGKY chain, but other framework lies behind. This method is developed as for the neutral as for the charged particles, but in our presentation we consider charged particles only and we consider non-relativistic limit. However, this method can be used for derivation of equations for relativistic hydrodynamic and kinetic as well. We demonstrate this method, as it has written, for simplest case. One of the interesting consequences is that electric dipole fleld, electric quadrupole fleld and etc are arise in equations of collective motion and in Maxwell equation as sources of electric fleld.