v == (PI' P2,"" P3N; q1' q2,"" q3N) and n the vector locally normal to the surface S, then

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d -- f dw fs dS n dt

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With the help of the divergence theorem in 6N-dimensional space, we convert this to the equation

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f dW[aat +

(vp )]

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(3.40)

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where \7 is the 6N-dimensional gradient operator:

(a: 1 ' a:2 "'" a:3N ; a:1' a: 2"'"

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Since w is an arbitrary volume the integrand of (3.40) must identically vanish. Hence

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3N a a] L [ -(PiP) + -(qiP) ;=1 api aqi p ) 3N (a p . aq ) 3N ap a L ( -P + -q. + L p - ' + - ' ;~1 api' aqi' ;=1 api aqi

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By the equations of motion (3.38) we have

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Therefore

(i = 1, ... ,3N)

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Liouville's theorem is equivalent to the statement (3.41)

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since by virtue of the equations of motion Pi and q; are functions of the time. Its

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THE PROBLEM OF KINETIC THEORY

geometrical interpretation is as follows. If we follow the motion of a representative point in r space, we find that the density of representative points in its neighborhood is constant. Hence the distribution of representative points moves in r space like an incompressible fluid. The observed value of a dynamical quantity of the system, which is generally a function of the coordinates and conjugate momenta, is supposed to be its averaged value taken over a suitably chosen ensemble:

pd 3NqO(p, q)p(p, q, t)

= --------

(3.42)

p d q p( p, q, t)

This is called the ensemble average of 0. Its time dependence comes from that of p, which is governed by Liouville's theorem. In principle, then, this tells us how a quantity approaches equilibrium-the central question of kinetic theory. In the next section we shall derive the Boltzmann transport equation using this approach. Under certain conditions one can prove an ergodic theorem, which says that if one waits a sufficiently long time, the locus of the representative point of a system will cover the entire accessible phase space. More precisely, it says that the representative point comes arbitrarily close to any point in the accessible phase space. This would indicate that the ensemble corresponding to thermodynamic equilibrium is one for which p is constant over the accessible phase space. This is actually what we shall assume. *

3.5 THE BBGKY HIERARCHY One can define correlation functions fs' which give the probability of finding s particles having specified positions and momenta, in the systems forming an ensemble. The function f1 is the familiar distribution function. The exact equations of motion for fs in classical mechanics can be written down. They show that to find f1 we need to know f2' which in turns depends on a knowledge of f3' and so on till we come the full N-body correlation function f N' This system of equations is known as the BBGKyt hierarchy. We shall derive it and show how the chain of equations can be truncated to yield the Boltzmann transport equation. The "derivation" will not be any more rigorous than the one already given, but it will give new insight into the nature of the approximations. Consider an ensemble of systems, each being a gas of N molecules enclosed m volume V, with Hamiltonian .Yr. Instead of the general notation {Pi' qi}

*See the remarks about the relevance of the ergodic theorem in Section 4.5. tBBGKY stands for Bogoliubov-Bom-Green-Kirkwood-Yvon. For a detailed discussion and references see N. N. Bogoliubov in Studies in Statistical Mechanics, J. de Boer and G. E. Uhlenbeck, Eds., Vol. I (North-Holland, Amsterdam, 1962).

THERMODYNAMICS AND KINETIC THEORY

(i = 1, (i = 1,

, 3N), we shall denote the coordinates by the Cartesian vectors {Pi' r i } , N), for which we use the abbreviation (3.43)

The density function characterizing the ensemble is denoted by p(l, ... , N, t), and assumed to be symmetric in Z1, . ' ZN. Its integral over all phase space is a constant by Liouville's theorem; hence we can normalized it to unity:

f dZ

(0) ==

1 .

dz N P(1, ... , N, t) = 1

(3.44)