# Hydrodynamic approximation

A method for the description of the evolution of a system and its typical properties in terms of the macroscopic equations of hydrodynamics. In a hydrodynamic approximation a system of the type of a gas or a liquid is considered as a continuous medium; any increments in the time $t$ (including $dt$) in the equations of hydrodynamics describing the system being always larger than the time of relaxation towards a distribution of local equilibrium (for a classical system — towards the local Maxwell distribution), i.e. being always larger than the times of formation of the local hydrodynamic characteristics such as density, thermodynamic velocity, temperature, etc., while the changes of the latter in the coordinate space are so smooth that the approximate volume increments $d\mathbf{r}$ do not merely contain a sufficient number of particles but also form quasi-homogeneous statistical systems.

From the point of view of statistical mechanics, the classical equations of hydrodynamics may be obtained from the kinetic equation in an approximation of slow and smoothed processes on a molecular scale (average free path) of time and length. To do this, the kinetic equation is employed to construct local density equations (continuity equations), hydrodynamic velocity equations (equations of motion) and local temperature equations (equations of conservation of energy); the solution for a single-particle distribution function, corresponding to the case of a small deviation from a local Maxwell distribution, is then substituted into these equations. In the zero-th approximation this yields an ideal liquid; in the first approximation the Navier–Stokes equations are obtained. This process forms the base of the Chapman–Enskog method.

A more general method is based on a chain of equations for time correlation functions (cf. Bogolyubov chain of equations) and its solution by expanding in the parameter characterizing the inhomogeneities of the system. The resulting expansions for the transport coefficients are identical with the results obtained by the Chapman–Enskog method only in those parts comprising collisions between two particles (triple collisions make a comparable contribution).

A similar method of obtaining hydrodynamic equations may also be applied to quantum liquids, when the starting point must be the Bogolyubov chain of equations for quantum correlation functions or the equations for the quantum Green function or the Schrödinger equation directly.

#### References

[1] | N.N. Bogolyubov, "Selected works" , 2 , Kiev (1970) (In Russian) |

[2] | G.E. Uhlenbeck, G.V. Ford, "Lectures in statistical mechanics" , Amer. Math. Soc. (1963) |

#### Comments

The Bogolyubov chain of equations is better known in the Western world as the BBGKY-hierarchy of equations.

#### References

[a1] | T.D. Cowling, "The mathematical theory of nonuniform gases" , Cambridge Univ. Press (1952) |

[a2] | C. Cercignani, "Mathematical methods in kinetic theory" , Plenum (1969) |

[a3] | C. Cercignani, "The Boltzmann equation and its applications" , Springer (1988) |

**How to Cite This Entry:**

Hydrodynamic approximation.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Hydrodynamic_approximation&oldid=33593