In
statistical physics
Statistical physics is a branch of physics that evolved from a foundation of statistical mechanics, which uses methods of probability theory and statistics, and particularly the Mathematics, mathematical tools for dealing with large populations ...
, the BBGKY hierarchy (Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy, sometimes called Bogoliubov hierarchy) is a set of equations describing the dynamics of a system of a large number of interacting particles. The equation for an ''s''-particle
distribution function (probability density function) in the BBGKY hierarchy includes the (''s'' + 1)-particle distribution function, thus forming a coupled chain of equations. This formal theoretic result is named after
Nikolay Bogolyubov
Nikolay Nikolayevich Bogolyubov (russian: Никола́й Никола́евич Боголю́бов; 21 August 1909 – 13 February 1992), also transliterated as Bogoliubov and Bogolubov, was a Soviet and Russian mathematician and theoretica ...
,
Max Born
Max Born (; 11 December 1882 – 5 January 1970) was a German physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a n ...
,
Herbert S. Green
Herbert Sydney Green (17 December 1920 – 16 February 1999) was a British–Australian physicist. Green was a doctoral student of the Nobel Laureate Max Born at Edinburgh, with whom he was involved in the development of the modern kinetic theor ...
,
John Gamble Kirkwood
John "Jack" Gamble Kirkwood (May 30, 1907, Gotebo, Oklahoma – August 9, 1959, New Haven, Connecticut) was a noted chemist and physicist, holding faculty positions at Cornell University, the University of Chicago, California Institute of Technol ...
, and
Jacques Yvon
Ancient and noble French family names, Jacques, Jacq, or James are believed to originate from the Middle Ages in the historic northwest Brittany region in France, and have since spread around the world over the centuries. To date, there are over ...
.
Formulation
The evolution of an ''N''-particle system in absence of
quantum fluctuations
In quantum physics, a quantum fluctuation (also known as a vacuum state fluctuation or vacuum fluctuation) is the temporary random change in the amount of energy in a point in space, as prescribed by Werner Heisenberg's uncertainty principle. ...
is given by the
Liouville equation
:''For Liouville's equation in dynamical systems, see Liouville's theorem (Hamiltonian).''
: ''For Liouville's equation in quantum mechanics, see Von Neumann equation.''
: ''For Liouville's equation in Euclidean space, see Liouville–Bratu–Gel ...
for the probability density function
in 6''N''-dimensional phase space (3 space and 3 momentum coordinates per particle)
:
where
are the coordinates and momentum for
-th particle with mass
, and the net force acting on the
-th particle is
:
where
is the pair potential for interaction between particles, and
is the external-field potential. By integration over part of the variables, the Liouville equation can be transformed into a chain of equations where the first equation connects the evolution of one-particle probability density function with the two-particle probability density function, second equation connects the two-particle probability density function with the three-particle probability density function, and generally the ''s''-th equation connects the ''s''-particle probability density function
:
with the (''s'' + 1)-particle probability density function:
:
The equation above for ''s''-particle distribution function is obtained by integration of the Liouville equation over the variables
. The problem with the above equation is that it is not closed. To solve
, one has to know
, which in turn demands to solve
and all the way back to the full Liouville equation. However, one can solve
, if
could be modeled. One such case is the
Boltzmann equation
The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium, devised by Ludwig Boltzmann in 1872.Encyclopaedia of Physics (2nd Edition), R. G. Lerne ...
for
, where
is modeled based on the
molecular chaos hypothesis (). In fact, in the Boltzmann equation
is the collision integral. This limiting process of obtaining Boltzmann equation from Liouville equation is known as
Boltzmann–Grad limit.
Physical interpretation and applications
Schematically, the Liouville equation gives us the time evolution for the whole
-particle system in the form
, which expresses an incompressible flow of the probability density in phase space. We then define the reduced distribution functions incrementally by integrating out another particle's degrees of freedom
. An equation in the BBGKY hierarchy tells us that the time evolution for such a
is consequently given by a Liouville-like equation, but with a correction term that represents force-influence of the
suppressed particles
:
The problem of solving the BBGKY hierarchy of equations is as hard as solving the original Liouville equation, but approximations for the BBGKY hierarchy (which allow truncation of the chain into a finite system of equations) can readily be made. The merit of these equations is that the higher distribution functions
affect the time evolution of
only implicitly via
Truncation of the BBGKY chain is a common starting point for many applications of kinetic theory that can be used for derivation of classical
or quantum kinetic equations. In particular, truncation at the first equation or the first two equations can be used to derive classical and quantum
Boltzmann equation
The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium, devised by Ludwig Boltzmann in 1872.Encyclopaedia of Physics (2nd Edition), R. G. Lerne ...
s and the first order corrections to the Boltzmann equations. Other approximations, such as the assumption that the density probability function depends only on the relative distance between the particles or the assumption of the hydrodynamic regime, can also render the BBGKY chain accessible to solution.
Bibliography
''s''-particle distribution functions were introduced in classical statistical mechanics by J. Yvon in 1935. The BBGKY hierarchy of equations for ''s''-particle distribution functions was written out and applied to the derivation of kinetic equations by Bogoliubov in the article received in July 1945 and published in 1946 in Russian
and in English.
The kinetic transport theory was considered by Kirkwood in the article received in October 1945 and published in March 1946, and in the subsequent articles. The first article by Born and Green considered a general kinetic theory of liquids and was received in February 1946 and published on 31 December 1946.
See also
*
Fokker–Planck equation
In statistical mechanics, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as ...
*
Vlasov equation The Vlasov equation is a differential equation describing time evolution of the Distribution function (physics), distribution function of plasma (physics), plasma consisting of charged particles with long-range interaction, e.g. Coulomb's law, Coulo ...
*
Cluster-expansion approach
The cluster-expansion approach is a technique in quantum mechanics that systematically truncates the BBGKY hierarchy problem that arises when quantum dynamics of interacting systems is solved. This method is well suited for producing a closed set ...
References
{{Statistical mechanics topics
Statistical mechanics
Non-equilibrium thermodynamics
Max Born