The ratio of the average microscopic agitation interval to the macroscopic relaxation time is proposed as the expansion parameter of linear Boltzmann or master operators. This parameter is interpretable physically both as a measure of the discontinuity of the random process, and as an inverse measure of the size of the fluctuating system. In the limit when the expansion parameter is zero, the process becomes continuous and is described by the Fokker‐Planck equation. When the parameter is nonvanishing, the expansion of the master operator in terms of it is, in three representative cases, a ``CD expansion'' in products of creation and destruction operators for Hermite functions; the dominant term is usually the Fokker‐Planck operator. These results are considered in relation to van Kampen's hypothesis for small‐parameter expansions of the same operators. It is found that the CD expansion fits the available model processes exactly, and that these processes do not satisfy van Kampen's hypothesis. As a new application, the explicit CD series is given for the density fluctuation model. Special cases of the model include the density fluctuations studied by van Kampen and the Ehrenfest urn model.

Topics
Operator theory, Stochastic processes
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The word “hypothesis” is not intended to convey the impression of a proposition supposed to be universally true. Rather we wish to suggest that it is true in certain ideal cases which are, so to speak, central to this subject. Compare with the use of the word “law” in the “ideal gas law,” or “law of large numbers,” neither of which is universally valid.
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This is almost the expression that van Kampen gives on p. 566 of his article.12 However, it is extremely important to note that there is a misprint in the article: Ω−1/2 should be replaced by Ω−ν/2 as in Eq. (2.28).
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33.
One may proceed as follows: The transition probability for the Maxwell molecule is obtained from the Rayleigh model by omission of the factor |v′−v|/2M1, which is |x| in the notation of Siegel’s paper. This means that the following changes would have to be made in his formulas to get αnk for Maxwellian molecules (numbers refer to equations in Siegel’s paper): In the quantity J, Eq. (B10), n is to be replaced by n−1; and in the exponent of μ/(l+μ) in Eq. (B13), k−n−2 is to be replaced by k−n−1. Note that the definition of αnk in Siegel’s paper differs from that in the present paper.
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The integral in (7.9), as well as that in (7.10), is to be evaluated by first substituting 1/ξ2 = −12 ∫ −∞∞|u|e−iuξdu, and then integrating with respect first to ξ and then to u. The result is convergent.
36.
This definition of the expansion parameter may appear unsuitable in view of the assertion that a is an extensive variable and Ω is the “size” of the system since a valid expansion parameter must be dimensionless (otherwise it has a “magnitude” which depends on the units used, and cannot ever be said to be “small”). However, when the size of the system is definable, it is usually possible to express it as a ratio to some fixed physical quantity of the same dimensionality, thereby making it dimensionless.
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1965
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