BBGKY hierarchy

In statistical physics, 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 Bogoliubov, Born, Green, Kirkwood, and Yvon.

Formulation

The evolution of an N-particle system in absence of quantum fluctuations is given by the Liouville equation for the probability density function $f_{N}=f_{N}({\mathbf {q}}_{1}\dots {\mathbf {q}}_{N},{\mathbf {p}}_{1}\dots {\mathbf {p}}_{N},t)$ in 6N dimensional phase space (3 space and 3 momentum coordinates per particle)

{\frac {\partial f_{N}}{\partial t}}+\sum _{{i=1}}^{N}{\dot {{\mathbf {q}}}}_{i}{\frac {\partial f_{N}}{\partial {\mathbf {q}}_{i}}}+\sum _{{i=1}}^{N}\left(-{\frac {\partial \Phi _{i}^{{ext}}}{\partial {\mathbf {q}}_{i}}}-\sum _{{j=1}}^{N}{\frac {\partial \Phi _{{ij}}}{\partial {\mathbf {q}}_{i}}}\right){\frac {\partial f_{N}}{\partial {\mathbf {p}}_{i}}}=0.

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 $f_{s}=f_{s}({\mathbf {q}}_{1}\dots {\mathbf {q}}_{s},{\mathbf {p}}_{1}\dots {\mathbf {p}}_{s},t)$ with the (s+1)-particle probability density function:

{\frac {\partial f_{s}}{\partial t}}+\sum _{i=1}^{s}{\dot {\mathbf {q} }}_{i}{\frac {\partial f_{s}}{\partial \mathbf {q} _{i}}}+\sum _{i=1}^{s}\left(-{\frac {\partial \Phi _{i}^{ext}}{\partial \mathbf {q} _{i}}}-\sum _{j=1}^{s}{\frac {\partial \Phi _{ij}}{\partial \mathbf {q} _{i}}}\right){\frac {\partial f_{s}}{\partial \mathbf {p} _{i}}}=(N-s)\sum _{i=1}^{s}{\frac {\partial }{\partial \mathbf {p} _{i}}}\int {\frac {\partial \Phi _{i\,s+1}}{\partial \mathbf {q} _{i}}}\cdot f_{s+1}\,d\mathbf {q} _{s+1}d\mathbf {p} _{s+1}.

Here ${\mathbf {q}}_{i},{\mathbf {p}}_{i}$ are the coordinates and momentum for ith particle, $\Phi ^{{ext}}({\mathbf {q}}_{i})$ is the external field potential, and $\Phi _{{ij}}({\mathbf {q}}_{i},{\mathbf {q}}_{j})$ is the pair potential for interaction between particles. The equation above for s-particle distribution function is obtained by integration of the Liouville equation over the variables ${\mathbf {q}}_{{s+1}}\dots {\mathbf {q}}_{N},{\mathbf {p}}_{{s+1}}\dots {\mathbf {p}}_{N}$ .

Physical interpretation and applications

Schematically, the Liouville equation gives us the time evolution for the whole $N$ -particle system in the form $Df_{N}=0$ , 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 $f_{s}\sim \int f_{{s+1}}$ . An equation in the BBGKY hierarchy tells us that the time evolution for such a $f_{s}$ is consequently given by a Liouville-like equation, but with a correction term that represents force-influence of the N-s suppressed particles

Df_{s}\propto {\text{div}}_{{{\mathbf p}}}\langle {\text{grad}}_{{{\mathbf q}}}\Phi _{{i,s+1}}\rangle _{{f_{{s+1}}}}.

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 $f_{{s+2}},f_{{s+3}},\dots$ affect the time evolution of $f_{s}$ only implicitly via $f_{{s+1}}.$ Truncation of the BBGKY chain is a common starting point for many applications of kinetic theory that can be used for derivation of classical^[1]^[2] or quantum^[3] kinetic equations. In particular, truncation at the first equation or the first two equations can be used to derive classical and quantum Boltzmann equations 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.^[4] 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 paper received on July, 1945 and published in 1946 in Russian^[1] and in English.^[2] The kinetic transport theory was considered by Kirkwood in the paper^[5] received on October, 1945 and published on March, 1946, and in the subsequent papers.^[6] The first paper by Born and Green considered a general kinetic theory of liquids and was received on February, 1946 and published on 31 December 1946.^[7]

References

1 2 N. N. Bogoliubov (1946). "Kinetic Equations". Journal of Experimental and Theoretical Physics (in Russian). 16 (8): 691–702.
1 2 N. N. Bogoliubov (1946). "Kinetic Equations". Journal of Physics USSR. 10 (3): 265–274.
↑ N. N. Bogoliubov, K. P. Gurov (1947). "Kinetic Equations in Quantum Mechanics". Journal of Experimental and Theoretical Physics (in Russian). 17 (7): 614–628.
↑ J. Yvon (1935): La théorie statistique des fluides et l'équation d'état (in French), Actual. Sci. & Indust. № 203 (Paris, Hermann).
↑ John G. Kirkwood (March 1946). "The Statistical Mechanical Theory of Transport Processes I. General Theory". The Journal of Chemical Physics. 14 (3): 180. Bibcode:1946JChPh..14..180K. doi:10.1063/1.1724117.
↑ John G. Kirkwood (January 1947). "The Statistical Mechanical Theory of Transport Processes II. Transport in Gases". The Journal of Chemical Physics. 15 (1): 72. Bibcode:1947JChPh..15...72K. doi:10.1063/1.1746292.
↑ M. Born and H. S. Green (31 December 1946). "A General Kinetic Theory of Liquids I. The Molecular Distribution Functions". Proc. Roy. Soc. A. 188: 10–18. Bibcode:1946RSPSA.188...10B. doi:10.1098/rspa.1946.0093.

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BBGKY hierarchy

Formulation

Physical interpretation and applications

Bibliography

See also

References