BBGKY (Bogolyubov, Born, Green, Kirkwood
and Yvon) hierarchy is a chain of an infinite number of integrodifferential
evolution equations for the sequence of distribution functions of particle
clustersdescribing all possible states of many-particle systems. For
the system of finite number of particles the BBGKY hierarchy is an equivalent
to the Liouville equation in the classical case and the von Neumann
equation in quantum case.
The BBGKY hierarchy describes both equilibrium
and non-equilibrium states of many-particle systems from a common
point of view. Non-equilibrium states are characterized by the solutions
of initial value problem for this hierarchy and, correspondingly,
equilibrium states are characterized by the solutions of the stationary
BBGKY hierarchy, see [1-3].
In the so-called large scale limits
asymptotics of solutions to the initial value problem of the BBGKY
hierarchy are governed by nonlinear kinetic equations or by the hydrodynamic
equations dependig on initial data. For example, in the Boltzmann-Grad
limit the asymptotics of the BBGKY hierarchy solutions is described
by the Boltzmann hierarchy and, as a consequence, for factorized initial
data the equation determining the evolution of such initial state
is the Boltzmann equation.
General references for this area are
See also: Liouville equation, Kinetic
equations, Gibbs states, Boltzmann-Grad limit.
Cercignani, C., Gerasimenko,
V., Petrina, D.: Many-Particle Dynamics and Kinetic Equations,
Kluwer Acad. Publ., 1997.
Petrina, D., Gerasimenko,
V., Malyshev, P.: Mathematical Foundations of Classical Statistical
Mechanics. Continuous systems, Second ed.: Taylor &
Petrina, D.: Mathematical
Foundations of Quantum Statistical Mechanics, Kluwer Acad.
Cercignani, C., Illner,
R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases,
Spohn, H.: Large
Scale Dynamics of Interacting Particles, Springer, 1991.