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''Bogolyubov–Born–Green–Kirkwood–Yvon hierarchy''

A chain of an infinite number of integro-differential evolution equations for the sequence of distribution (correlation) functions describing all possible states of a many-particle system. For systems of finitely many particles the BBGKY hierarchy is equivalent to the [[Liouville-equation(2)|Liouville equation]] in the classical case and to the von Neumann equation in the quantum case.

Let $F ( t ) = ( F _ { 1 } ( t , x _ { 1 } ) , \ldots , F _ { n } ( t , x _ { 1 } , \ldots , x _ { n } ) , \ldots )$ be a sequence of probability distribution functions $F _ { n } ( t )$ defined on the phase space of a system of $n$ particles $x _ { i } \equiv ( q _ { i } , p _ { i } ) \in \mathbf{R} ^ { \nu } \times \mathbf{R} ^ { \nu }$, $\nu \geq 1$ (cf. also [[Distribution function|Distribution function]]). The BBGKY hierarchy for classical systems reads as follows:

\begin{equation*} \frac { d } { d t } F ( t ) = - \mathcal{L} F ( t ) + [ \mathcal{L} , A ] F ( t ), \end{equation*}

where the Liouville operator $\mathcal{L}$ is defined by the [[Poisson brackets|Poisson brackets]] ($X \equiv ( x _ { 1 } , \dots , x _ { n } )$):

\begin{equation*} ( \mathcal{L} F ) _ { n } ( X ) = \{ H _ { n } , F _ { n } ( X ) \} \end{equation*}

with [[Hamiltonian|Hamiltonian]] $H _ { n } = \sum _ { i = 1 } ^ { n } p _ { i } ^ { 2 } / 2 + \sum _ { 1 = i < j } ^ { n } \Phi ( q _ { i } - q _ { j } )$ and the symbol $[ \cdot , \cdot ]$ denotes the commutator between the operator $\mathcal{L}$ and the operator

\begin{equation*} ( \mathcal{A} F ) _ { n } ( X ) = \int d x _ { n  + 1} F _ { n  + 1} ( X , x _ { n  + 1} ). \end{equation*}

The main examples of the BBGKY hierarchy are discussed below [[#References|[a1]]].

1) For a smooth [[Potential|potential]] of interaction $\Phi$,

\begin{equation*} ( [ {\cal L , A} ] F ) _ { n } ( X ) = \end{equation*}

<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120100/b12010016.png"/></td> </tr></table>

2) For a system with singular potential of interaction

\begin{equation*} \Phi ( q ) = \left\{ \begin{array} { l l } { + \infty } &amp; { \text { if } | q | \leq \sigma , } \\ { 0 } &amp; { \text { if } | q | > \sigma , } \end{array} \right. \end{equation*}

like a hard-sphere system of diameter $\sigma$, for dimension $\nu \geq 2$ at $t > 0$ one has

\begin{equation*} ( [ {\cal L , A} ] F ) _ { n } ( X ) = \end{equation*}

<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120100/b12010022.png"/></td> </tr></table>

\begin{equation*} - F _ { n + 1 } ( X , q _ { i } + \sigma \eta , p _ { n + 1 } ) ), \end{equation*}

where $\langle \, .\, ,\,  . \, \rangle$ is the inner product,

\begin{equation*} \mathcal{S} _ { + } ^ { \nu - 1 } = \left\{ \eta \in \mathbf{R} ^ { \nu } : | \eta | = 1 , \langle \eta , ( p _ { i } - p _ { n + 1 } ) \rangle > 0 \right\} \end{equation*}

and

\begin{equation*} p _ { i } ^ { * } = p _ { i } - \eta \langle \eta , ( p _ { i } - p _ { n + 1 } ) \rangle, \end{equation*}

<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120100/b12010027.png"/></td> </tr></table>

3) The dual BBGKY hierarchy

\begin{equation*} \frac { d } { d t } G ( t ) = \mathcal{L} G ( t ) + [ \mathcal{L} , \mathcal{A} ^ { * } ] G ( t ), \end{equation*}

where $\mathcal{A} ^ { * }$ is the [[Adjoint operator|adjoint operator]] to $\mathcal{A}$:

\begin{equation*} ( \mathcal{A} ^ { * } f ) _ { n } ( X ) = \sum _ { i = 1 } ^ { n } f _ { n - 1 } ( x _ { 1 } , \dots , x _ { i  - 1} , x _ { i  + 1} , \dots , x _ { n } ). \end{equation*}

Such a hierarchy describes the evolution of observables $G ( t )$ of a system of particle.

There are two different approaches to constructing the solutions of the hierarchy: either by iteration or by a functional (cluster expansion) series; see [[#References|[a1]]]. The functional representation of the solution for the Cauchy problem depends on symmetry properties of the Hamiltonian and on initial distribution functions

\begin{equation*} F ( 0 ) = ( F _ { 1 } ( 0 , x _ { 1 } ) , \ldots , F _ { n } ( 0 , x _ { 1 } , \ldots , x _ { n } ) , \ldots ). \end{equation*}

In the symmetric case it has the form

\begin{equation*} F _ { s } ( t , x _ { 1 } , \ldots , x _ { s } ) = \end{equation*}

\begin{equation*} = \sum _ { n = 0 } ^ { \infty } \int d x _ { s + 1} \cdots d x _ { s + n} U ^ { ( n )_t }  F _ { s + n} ( 0 , x _ { 1 } , \dots , x _ { s  + n} ), \end{equation*}

where

\begin{equation*} U ^ { ( n )_ t} = \sum _ { k = 0 } ^ { n } \frac { ( - 1 ) ^ { k } } { k ! ( n - k ) ! } S ^ { s + n - k } ( - t , x _ { 1 } , \dots , x _ { s  + n - k} ) \end{equation*}

and the evolution operator $S ^ { n } ( - t , x _ { 1 } , \dots , x _ { n } )$,

\begin{equation*} S ^ { n } ( - t , x _ { 1 } , \dots , x _ { n } ) F _ { n } ( x _ { 1 } , \dots , x _ { n } ) = \end{equation*}

\begin{equation*} = F _ { n } ( X _ { 1 } ( - t , x _ { 1 } , \ldots , x _ { n } ) , \ldots , X _ { n } ( - t , x _ { 1 } , \ldots , x _ { n } ) ), \end{equation*}

shifts the arguments of the functions along the solutions of the Hamilton equations (the phase trajectories) $X _ { i } ( - t , x _ { 1 } , \ldots , x _ { n } )$, $i = 1 , \dots , n$, of a system of $n$ particles with initial data $X _ { i } ( 0 , x _ { i } ) = x _ { i }$. The evolution operators $S ^ { n } ( t )$ form a group of operators generated by the Liouville operator ${\cal L} _ { n }$ defined above. In certain cases this representation of the solution (a cluster expansion) can be treated as the solution $F ( t ) = U ( t ) F ( 0 )$ determined by a group of evolution operators $U ( t )$. In terms of the operators defined above, the group $U ( t )$ has the form

\begin{equation*} U ( t ) = e ^ { \mathcal{A} } S ( - t ) e ^ { - \mathcal{A} }. \end{equation*}

There exist various methods to give a meaning to the formal solution of the Cauchy problem to the BBGKY hierarchy in certain function spaces. For example, the semi-group method (in spaces of sequences of summable functions and, for quantum systems, in the space of sequences of kernel operators), the interaction region method (for constructing an a priori estimate of a solution), the thermodynamic limit method (for the time-continuation of the local solution in the space of sequences of bounded functions), and non-standard analysis methods.

To date (1998), existence and uniqueness results for the BBGKY hierarchy have been proved:

a) in the space $L^1$ of sequences of summable functions (or kernel operators, in the quantum case): time-global solutions for general classes of an interaction potential;

b) in the space $L^{\infty}$ of sequences of functions bounded with respect to the configuration variables and decreasing with respect to the momentum variables: time-global solutions for the BBGKY hierarchy of a hard-sphere system for arbitrary initial data in the one-dimensional case and with initial data close to the equilibrium state in the many-dimensional case; also, time-global solutions for one-dimensional systems of particles interacting by the short-range potential with a hard-core and with initial data close to equilibrium.

The BBGKY hierarchy allows one to describe 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 the initial value problem for this hierarchy and, correspondingly, equilibrium states are characterized by the solutions of the steady BBGKY hierarchy, see [[#References|[a1]]], [[#References|[a3]]], [[#References|[a2]]].

In so-called large-scale limits, asymptotics of solutions to the initial value problem of the BBGKY hierarchy are governed by non-linear kinetic equations (cf. also [[Kinetic equation|Kinetic equation]]) or by hydrodynamic equations depending on initial data. For example, in the [[Boltzmann–Grad limit|Boltzmann–Grad limit]] the asymptotics of solutions of the BBGKY hierarchy is described by the Boltzmann hierarchy and, as a consequence, for factorized initial data the equation determining the evolution of an initial state is the [[Boltzmann equation|Boltzmann equation]].

General references for this area are [[#References|[a1]]], [[#References|[a4]]], [[#References|[a3]]], [[#References|[a2]]], [[#References|[a5]]].

====References====
<table>
<tr><td valign="top">[a1]</td> <td valign="top">  C. Cercignani,   V. Gerasimenko,   D. Petrina,   "Many-particle dynamics and kinetic equations" , Kluwer Acad. Publ.  (1997)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  D. Petrina,   V. Gerasimenko,   P. Malyshev,   "Mathematical foundations of classical statistical mechanics. Continuous systems" , Gordon&amp;Breach  (1989)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  D. Petrina,   "Mathematical foundations of quantum statistical mechanics" , Kluwer Acad. Publ.  (1995)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  C. Cercignani,   R. Illner,   M. Pulvirenti,   "The mathematical theory of dilute gases" , Springer  (1994)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  H. Spohn,   "Large scale dynamics of interacting particles" , Springer  (1991)</td></tr>
</table>