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The transport phenomena in gases---viscosity, diffusion, and the conduction of heat and electricity---have considerable interest in problems of the sun and stars, the earth's upper atmosphere and magnetosphere, and the solar wind.
When Chapman began his fundamental work on these phenomena a little over fifty years ago, most discussions of them employed a crude free-path argument. Boltzmann indeed had given his celebrated equation for the determination of the velocity-distribution function many years before, but this had in the main been used only in discussions of equilibrium properties; Maxwell had likewise given his equations of transfer of molecular properties, but these had led to exact expressions for the transport coefficients only for molecules repelling each other with a force proportional to the inverse fifth power of the distance---the so-called Maxwellian molecules. The only other case where the transport coefficients had been calculated was that of the Lorentz approximation---a mixture of heavy and very light molecules, in which the collisions between the light molecules are of negligible importance.
The problem of the transport phenomena presented a double challenge. It was a challenge to mathematicians, because an integral equation had to be solved (integral equations were then just becoming popular). On the other hand, physicists needed formulas from which to calculate values of the transport coefficients, which could be verified experimentally. Chapman accepted the mathematical challenge, but never lost sight of the physicist's needs.
Enskog, improving on a suggestion by Hilbert, was working on the problem at the same time as Chapman. Their results were very similar; each led to a method of calculating the transport coefficients to any desired order of approximation. The contrast between their methods is, however, interesting. Enskog sought a well-posed mathematical problem, which he then solved by a series expansion, determining the coefficients by a minimum principle; he proceeded from a general formal solution of an integral equation to special approximate results. Chapman was less interested in the mathematical niceties, feeling that Nature herself had shown that a solution to the problem must exist. From the start he sought to construct an approximate solution, beginning with the simplest possible approximations and proceeding to expressions of steadily increasing complexity. Whereas Enskog emphasized the Boltzmann equation, Chapman concentrated on Maxwell's equation of transfer; but their final results agreed. The two approaches were complementary; Enskog provided a fuller justification for their joint results, but Chapman was quick to exploit their physical significance.
One outcome of their work was the identification of a new phenomenon, that of thermal diffusion. Chapman has described his difficulties in getting the importance of this phenomenon accepted by the recognized authorities, notably J. H. Jeans, who thought that it must operate almost infinitely slowly. Chapman therefore encouraged F. W. Dootson to conduct experiments which clearly established the importance of the phenomenon. He has always felt pride in the discovery of thermal diffusion, and has often returned to analyze its properties in more detail. It gave him great pleasure when in 1939, K. Clusius, in his separation column, devised a powerful means of using thermal diffusion to effect gas separation.
Soon after World War I, Chapman decided to write a book on gas theory, embodying the joint work of Enskog and himself. He then wrote about one-third of what appears in The Mathematical Theory of Non-Uniform Gases, containing the essential framework, and introducing the vector and tensor notation used in the book. However, his many other interests prevented him from going further, and he sought a collaborator. He was not finally successful until 1931, and even after this the progress with the book was intermittent. It finally appeared in 1939.
In 1922, Chapman considered the transport phenomena in ionized gases, in the discussion of which the slow decrease of electrostatic forces with distance produces convergence difficulties. Much work has been done on ionized gases since then, but it is worth observing that, save as regards the important matter of the cut-off distance, remarkably little that is essentially new has been added to Chapman's work. He may not at the time have realized its full inwardness but, for example, his assumptions and results were virtually the same as those of the more recently popular Fokker-Planck approach. As with thermal diffusion, Chapman has returned often to the study of ionized gases, particularly in astronomical and geophysical contexts.
His success with a more exact theory led Chapman to be strongly critical of approximate theories of the transport phenomena, particularly of diffusion. As he stressed, such theories either fail to predict thermal diffusion, or are liable to give an incorrect expression for it. His objection was not leveled at approximate theories as such, but at theories whose physical basis was inadequate, or in which there was no provision for estimating the errors involved, say by calculating a better approximation. In his work on ionized gases in a magnetic field, for example, he has frequently used the approximate collision-interval theory employed by Appleton in his magneto-ionic work.
Of recent years Chapman's time has been largely engaged with matters other than classical kinetic theory, even though (e.g., in the discussion of pitch-angle distributions) kinetic theory is of great assistance in understanding many of these matters. Looking back on his work in this field, one must acknowledge that, first and foremost, he has been a trail-breaker. He not only reached important results himself; he laid a solid track for others to follow.
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