3.4 THE GIBBSIAN ENSEMBLE

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Gibbs introduced the idea of a statistical ensemble to describe a macroscopic system, which has proved to be a very important concept. We shall use it here to present another approach to the Boltzmann transport equation. A state of the gas under consideration can be specified by the 3N canonical coordinates ql' .. ' q3N and their conjugate momenta Pl, ., P3N. The 6Ndimensional space spanned by {Pi' qi} is called the r space, or phase space, of

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the system. A point in r space represents a state of the entire N-particle system, and is referred to as the representative point. This is in contrast to the I-t space introduced earlier, which refers to only one particle. It is obvious that a very large (in fact, infinite) number of states of the gas corresponds to a given macroscopic condition of the gas. For example, the condition that the gas is contained in a box of volume 1 cm3 is consistent with an infinite number of ways to distribute the molecules in space. Through macroscopic measurements we would not be able to distinguish between two gases existing in different states (thus corresponding to two distinct representative points) but satisfying the same macroscopic conditions. Thus when we speak of a gas under certain macroscopic conditions, we are in fact referring not to a single state, but to an infinite number of states. In other words, we refer not to a single system, but to a collection of systems, identical in composition and macroscopic condition but existing in different states. With Gibbs, we call such a collection of systems an ensemble, which is geometrically represented by a distribution of representative points in r space, usually a continuous distribution. It may be conveniently described by a density function p(p, q, t), where (p, q) is an abbreviation for (Pi' ... , P3N; ql' ... , q3N)' so defined that

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(3.37)

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is the number of representative points that at time t are contained in the infinitesimal volume element d 3Npd 3Nq of r space centered about the point (p, q). An ensemble is completely specified by p(p, q, t). It is to be emphasized that members of an ensemble are mental copies of a system and do not interact with one another. Given p(p, q, t) at any time t, its subsequent values are determined by the dynamics of molecular motion. Let the Hamiltonian of a system in the ensemble be .Yf(Pl, ... , P3N; ql' ., q3N) The equations of motion for a system are given by

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1, ... ,3N)

(3.38)

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

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These will tell us how a representative point moves in r space as time evolves. We assume that the Hamiltonian does not depend on any time derivative of p and q. It is then clear that (3.38) is invariant under time reversal and that (3.38) uniquely determines the motion of a representative point for all times, when the position of the representative point is given at any time. It follows immediately from these observations that the locus of a representative point is either a simple closed curve or a curve that never intersects itself. Furthermore, the loci of two distinct representative points never intersect. We now prove the following theorem.

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(3.39)

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Proof Since the total number of systems in an ensemble is conserved, the number of representative points leaving any volume in r space per second must be equal to the rate of decrease of the number of representative points in the same volume. Let w be an arbitrary volume in r space and let S be its surface. If we denote by v the 6N-dimensional vector whose components are

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