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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 [1-5].

See also: Liouville equation, Kinetic equations, Gibbs states, Boltzmann-Grad limit.


  1. Cercignani, C., Gerasimenko, V., Petrina, D.: Many-Particle Dynamics and Kinetic Equations, Kluwer Acad. Publ., 1997.
  2. Petrina, D., Gerasimenko, V., Malyshev, P.: Mathematical Foundations of Classical Statistical Mechanics. Continuous systems, Second ed.: Taylor & Francis, 2002.
  3. Petrina, D.: Mathematical Foundations of Quantum Statistical Mechanics, Kluwer Acad. Publ., 1995.
  4. Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases, Springer, 1994.
  5. Spohn, H.: Large Scale Dynamics of Interacting Particles, Springer, 1991.
Prof. Victor I. Gerasimenko
Institute of Mathematics, Kiev, Ukraine
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