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Thus H is at a local peak,t as illustrated in Fig. 4.5. When H is not at a local peak, such as at the points a and c in Fig. 4.5, the state of the gas is not a state of "molecular chaos." Hence molecular collisions, which are responsible for the change of H with time, can create "molecular chaos" when there is none and destroy "molecular chaos" once established. It is important to note that dH/ dt is not necessarily a continuous function of time; it can be changed abruptly by molecular collisions. Overlooking this fact might lead uS to conclude, erroneously, that the H theorem is inconsistent with the invariance under time reversal. A statement of the H theorem that is manifestly invariant under time reversal is the following. If there is "molecular chaos" now, then dH/dt:5: 0 in the next instant. If there will be "molecular chaos" in the next instant, then dH/dt ~ 0 now.

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*These simplifying features are introduced to avoid the irrelevant complications arising from the time reversal properties of the external force and the agent preparng the system. tThe foregoing argument is due to F. E. Low (unpublished).

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THE EQUILIBRIUM STATE OF A DILUTE GAS

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Fig. 4.5 H is at a local peak when the gas is in a state of "molecular chaos."

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We now discuss the general behavior of H as a function of time. Our discussion rests on the following premises.

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(a) H is at its smallest possible value when the distribution function is

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strictly Maxwell-Boltzmann. This easily follows from (4.50), and it is independent of the assumption of molecular chaos. * (b) Molecular collisions happen at random, i.e., the time sequence of the states of a gas is a sequence of states chosen at random from those that satisfy the macroscopic conditions. This assumption is plausible but unproved. From these premises it follows that the distribution function of the gas is almost always essentially Maxwell-Boltzmann, i.e., a distribution function contained within the peak shown in Fig. 4.4. The curve of H as a function of time consists mostly of microscopic fluctuations above the minimum value. Between two points at which H is at the minimum value there is likely to be a small peak. If at any instant the gas has a distribution function appreciably different from the Maxwell-Boltzmann distribution, then H is appreciably larger than the minimum value. Since collisions are assumed to happen at random, it is overwhelmingly probable that after the next collision the distribution will become essentially Maxwell-Boltzmann and H will decrease to essentially the minimum value. By time reversal invariance it is overwhelmingly probable that before the last collision H was at essentially the minimum value. Thus H is overwhelmingly likely to be at a sharp peak when the gas is in an improbable state. The more improbable the state, the sharper the peak. A very crude model of the curve of H as a function of time is shown in Fig. 4.6. The duration of a fluctuation, large or small, should be of the order of the time between two successive collisions of a molecule, i.e., 1O-1l sec for a gas under ordinary conditions. The large fluctuations, such as that labeled a in Fig. 4.6, almost never occur spontaneously.t We can, of course, prepare a gas in an improbable state, e.g., by suddenly removing a wall of the container of the gas, so that H is initially at a peak. But it is overwhelmingly probable that within a few collision times the distribution would be reduced to an essentially MaxwellBoltzmann distribution.

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*See Problem 4.9. tSee Problems 4.5 and 4.6.

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THERMODYNAMICS AND KINETIC THEORY

Fig. 4.6 H as a function of time. The range of values of H lying between the two horizontal dashed lines is called the" noise range."

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Most of the time the value of H fluctuates within a small range above the minimum value. This range, shown enclosed by dashed lines in Fig. 4.6, corresponds to states of the gas with distribution functions that are essentially Maxwell-Boltzmann, i.e., distribution functions contained within the peak of Fig. 4.4. We call this range the "noise range." These features of the curve of H have been deduced only through plausibility arguments, but they are in accord with experience. We can summarize them as follows.

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