THERMODYNAMICS AND KINETIC THEORY

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If the molecules are uniformly distributed in space, so that then

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f is independent of r,

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(3.5)

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jf(r,p, t) d p

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The aim of kinetic theory is to find the distribution function f(r, p, t) for a given form of molecular interaction. The limiting form of f(r, p, t) as t ~ 00 would then contain all the equilibrium properties of the system. The aim of kinetic theory therefore includes the derivation of the thermodynamics of a dilute gas. To fulfill this aim, our first task is to find the equation of motion for the distribution function. The distribution function changes with time, because molecules constantly enter and leave a given volume element in fL space. Suppose there were no molecular collisions (i.e., (J = 0). Then a molecule with the coordinates (r,p) at the instant t will have the coordinates (r + vSt,p + FSt) at the instant t + St, where F is the external force acting on a molecule, and v = p/m is the velocity. We may take St to be a truly infinitesimal quantity. Thus all the molecules contained in a fL-space element d 3 rd 3p, at (r, p), at the instant t, will all be found in an element d 3r' d 3p', at (r + v St, p + FSt), at the instant t + St. Hence in the absence of collisions we have the equality

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f(r + vSt,p + FSt, t + St) d 3 r' d3v'

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which reduces to

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f(r, v, t) d 3 rd 3u

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(3.6)

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f(r + v St, p + F St, t + St) = f(r, p, t)

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because d 3 rd 3p = d 3 r' d 3p'. The last fact is easily established if we assume that the external force F depends on position only. At any instant t, we may choose d 3 r d 3P to be a six-dimensional cube. It is sufficient to show that the area of any projection of this cube, say, dx dpx, does not change. A simple calculation will show that this projection, originally a square, becomes a parallelogram of the same area in the time St, as illustrated in Fig. 3.2. This invariance is valid as long as (r, p) are canonically conjugate generalized coordinates. When there are collisions (i.e., (J > 0), equality (3.6) must be modified. We write

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f f(r+vSt,p+FSt,t+St)=f(r,p,t)+(aa )

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(3.7)

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which defines ( a at) coli' Expanding the left-hand side to the first order in St, we f/ obtain the equation of motion for the distribution function as we let St ~ 0:

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f ( ~ + .! . V'r + F V'p)f(r,p, t) (aat ) at m

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(3.8)

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coli

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where V'r, V' p are, respectively, the gradient operators with respect to rand p. This equation is not meaningful until we explicitly specify (a f / at) coli' It is in specifying this term that the assumption that the system is a dilute gas becomes relevant.

THE PROBLEM OF KINETIC THEORY

+ at

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fir'x --------~ ~ at ~x

1 '---_1--+-----Jr--+---~ X

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Fig. 3.2 The invariance of the volume element in p, space under dynamical evolution in time.

An explicit form for (at/ a coli can be obtained by going back to its t) definition (3.7). Consider Fig. 3.3, where the square labeled A represents the JL-space volume element at {r, p, t} and the one labeled B represents that at {r + v 8t, P + F 8t, t + 8t}, where 8t eventually tends to zero. During the time interval 8t, some molecules in A will be removed from A by collision. We regard A as so small that any collision that a molecule in A suffers will knock it out of A. Such a molecule with not reach B. On the other hand, there are molecules outside A which, through collisions, will get into A during the time interval 8t. These will be in B. Therefore the number of molecules in Bat t + 8t, as 8t ~ 0, equals the original number of molecules in A at time t plus the net gain of molecules in A due to collisions during the time interval 8t. This statement is the content of (3.7), and may be expressed in the form