|
|
| Line 1: |
Line 1: |
| 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 [[Nikolay Bogolyubov|Bogoliubov]], [[Max Born|Born]], [[Herbert S. Green|Green]], [[John Gamble Kirkwood|Kirkwood]], and [[Jacques Yvon|Yvon]].
| | Hi there, I am Andrew Berryhill. North Carolina is exactly where we've been living for many years and will never move. What me and my family members love is bungee leaping but I've been using on new things recently. I am presently a journey agent.<br><br>Here is my web-site; free tarot readings [[http://help.ksu.edu.sa/node/65129 click the up coming internet site]] |
| | |
| ==Formulation==
| |
| The evolution of an ''N''-particle system is given by the [[Liouville's theorem (Hamiltonian)|Liouville equation]] for the probability density function <math>f_N = f_N(\mathbf{q}_1\dots\mathbf{q}_N, \mathbf{p}_1 \dots \mathbf{p}_N, t)</math> in ''6N'' phase space
| |
| :<math>
| |
| \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.
| |
| </math>
| |
| | |
| 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 <math>f_s = f_s(\mathbf{q}_1\dots\mathbf{q}_s, \mathbf{p}_1 \dots \mathbf{p}_s, t)</math> with the ''(s+1)''-particle probability density function:
| |
| :<math>
| |
| \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_{is+1}}{\partial \mathbf{q}_i}\cdot f_{s+1} \,d\mathbf{q}_{s+1} d\mathbf{p}_{s+1}.
| |
| </math>
| |
| Here <math>\mathbf{q}_i, \mathbf{p}_i</math> are the coordinates and momentum for ''i''th particle, <math>\Phi^{ext}(\mathbf{q}_i)</math> is the external field potential, and <math>\Phi_{ij}(\mathbf{q}_i, \mathbf{q}_j)</math> 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 <math>\mathbf{q}_{s+1}\dots\mathbf{q}_N, \mathbf{p}_{s+1} \dots \mathbf{p}_N</math>.
| |
| | |
| ==Physical interpretation and applications==
| |
| Schematically, the Liouville equation gives us the time evolution for the whole <math>N</math>-particle system in the form <math>Df_N=0</math>, 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 <math>f_s \sim \int f_{s+1}</math>. An equation in the BBGKY hierarchy tells us that the time evolution for such a <math>f_s</math> is consequently given by a Liouville-like equation, but with a correction term that represents force-influence of the N-s suppressed particles
| |
| | |
| :<math>D f_s \propto \text{div}_{\mathbf p} \langle \text{grad}_{\mathbf q}\Phi_{i,s+1}\rangle_{f_{s+1}}.</math>
| |
| | |
| 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 <math>f_{s+2},f_{s+3},\dots</math> affect the time evolution of <math>f_s</math> only implicitly via <math>f_{s+1}.</math> Truncation of the BBGKY chain is a common starting point for many applications of kinetic theory that can be used for derivation of classical<ref name="a">{{cite journal|author=[[Nikolay Bogoliubov|N. N. Bogoliubov]]|title=Kinetic Equations|journal=[[JETP|Journal of Experimental and Theoretical Physics]]|volume=16|issue=8|pages=691–702|year=1946|url=|language=Russian}}</ref><ref name="b">{{cite journal|author=[[Nikolay Bogoliubov|N. N. Bogoliubov]]|title=Kinetic Equations|journal=Journal of Physics USSR|volume=10|issue=3|pages=265–274|year=1946|url=}}</ref> or quantum<ref>{{cite journal|author=[[Nikolay Bogoliubov|N. N. Bogoliubov]], [[Kirill Gurov|K. P. Gurov]]|title=Kinetic Equations in Quantum Mechanics|journal=[[JETP|Journal of Experimental and Theoretical Physics]]|volume=17|issue=7|pages=614–628|year=1947|url=|language=Russian}}</ref> kinetic equations. In particular, truncation at the first equation or the first two equations can be used to derive classical and quantum [[Boltzmann equation]]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.<ref>[[Jacques Yvon|J. Yvon]] (1935): ''La théorie statistique des fluides et l'équation d'état'' (in French), Actual. Sci. & Indust. № 203 (Paris, Hermann).</ref> 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<ref name="a"/> and in English.<ref name="b"/> The kinetic transport theory was considered by Kirkwood in the paper<ref>{{cite journal|author=[[John Gamble Kirkwood|John G. Kirkwood]]|title=The Statistical Mechanical Theory of Transport Processes I. General Theory|journal=[[Journal of Chemical Physics|The Journal of Chemical Physics]]|volume=14|issue=3|pages=180|date=March 1946|doi=10.1063/1.1724117|bibcode = 1946JChPh..14..180K }}</ref> received on October, 1945 and published on March, 1946, and in the subsequent papers.<ref>{{cite journal|author=[[John Gamble Kirkwood|John G. Kirkwood]]|title=The Statistical Mechanical Theory of Transport Processes II. Transport in Gases|journal=[[Journal of Chemical Physics|The Journal of Chemical Physics]]|volume=15|issue=1|pages=72|date=January 1947|doi=10.1063/1.1746292 |bibcode = 1947JChPh..15...72K }}</ref> 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.<ref>{{cite journal|author=[[Max Born|M. Born]] and [[Herbert S. Green|H. S. Green]]|title=A General Kinetic Theory of Liquids I. The Molecular Distribution Functions|journal=Proc. Roy. Soc. A|volume=188|pages=10–18|date=31 December 1946|url=|bibcode = 1946RSPSA.188...10B |doi = 10.1098/rspa.1946.0093 }}</ref>
| |
| | |
| ==See also==
| |
| *[[Fokker–Planck equation]]
| |
| *[[Vlasov equation]]
| |
| | |
| ==References==
| |
| {{Reflist}}
| |
| | |
| {{DEFAULTSORT:Bbgky Hierarchy}}
| |
| <!--Categories-->
| |
| [[Category:Statistical mechanics]]
| |
| [[Category:Non-equilibrium thermodynamics]]
| |
Hi there, I am Andrew Berryhill. North Carolina is exactly where we've been living for many years and will never move. What me and my family members love is bungee leaping but I've been using on new things recently. I am presently a journey agent.
Here is my web-site; free tarot readings [click the up coming internet site]