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• W2085503627 abstract "This paper starts with a short review of the usual methods of stellar dynamics and analyzes the underlying hypotheses. The most successful approach to stellar dynamics has been the one which neglects interactions between nearby stars (because they are weak) and takes into consideration the motion of stars in a smooth field of force only, produced by the smeared-out mass disribution of the stellar system. This method is based on nothing but mechanics. It is, however, a semi-empirical one because it is confined to systems in such a state and for so short a duration that stellar encounters may be neglected. It does not explain how stellar systems reached their present state. This as well as the explanation of statistical distributions cannot be hoped to be achieved by merely using mechanics. It is necessary to introduce statistical hypotheses to provide the means for attacking problems of a statistical character, such as a theory of stellar systems including interactions of individual stars. A similar situation is exemplified by Clausius-Boltzmann's hypothesis concerning molecular chaos (Stosszahlansatz) which in its statistical interpretation provides the essential basis for deriving the second law of thermodynamics from the kinetic theory of gases. Compare with the discussion of the principles of kinetic theory by P. and T. Ehrenfest, Enzyclopaedie d. math. Wiss. 4, Art. 32, and R. C. Tolman, Principles of Statistical Mechanics. One attack on the general statistical problem is the method of force fluctuations. But this method is limited: In order to be able to calculate the relevant data about force fluctuations, it is necessary to assume the distances between, and the velocity- and mass-distributions of the stars, all numerical data about which have to be borrowed from observation. Besides, the actual interaction of nearby masses is a play between single stars, multiple stars, and star clusters, and it is impossible to calculate force fluctuations between those complex sub-systems of a stellar system without a great number of artificial assumptions. Therefore we should like to examine the consequence of a more basic hypothesis: Provided the large scale development of a stellar system is so slow that it can approximately be regarded as sliding from one statistically equilibrium state (as concerns the mutual adjustments of the elements) to another, we assume that statistically independent elements of a stellar system (e.g. galactic clusters or multiple stars or independent single stars as elements of their stellar system---our galaxy) cannot be crowded into Boltzmann phase space ${x}_{1}$, ${x}_{2}$, ${x}_{3}$, ${v}_{1}$, ${v}_{2}$, ${v}_{3}$ more closely than with an expectation value of one element per volume ${ensuremath{sigma}}^{3}$ where $ensuremath{sigma}$ is a constant, characteristic for the system. This is plausible because too great densities in position space, in the absence of large mean square deviations of velocities, give cause to an aggregation of formerly independent clusters or stars into larger statistically independent units. This assumption indicates the limits of determinacy of the mass distribution function $f({x}_{1},{x}_{2},{x}_{3},{v}_{1},{v}_{2},{v}_{3})$ if we try to evaluate $f$ on the basis of star counts in too small volumes in Boltzmann phase space. It also indicates the expected minimum distance between statistically independent elements in position space for a given mean square deviation of velocities. As we cannot expect a stellar system to be flattened into a disk more narrow than this minimum distance, a minimum relation between intervals in position space and intervals in velocity space follows, in mathematical analogy to the quantum-mechanical uncertainty principle. Our assumption is analogous to the Pauli principle (Thomas-Fermi atom). It is interesting to note that this exclusion principle forms the basic assumption in the astronomical case, and that the minimum relation is a consequence of it. Our hypothesis pays attention to microaspects of the streaming field and therefore implies an assumption about interaction of neighboring elements. (The word micro refers to the fine grained details of a streaming field and shall not imply that the independent elements are labeled and distinguished.) In the second part we investigate the transformation of the two classical hydrodynamical equations (continuity equation in position space, and Bernouilli equation) into a wave equation of the Schroedinger type and justify this procedure in the light of the foregoing remarks. The main idea is that we exploit the indeterminacy of the hydrodynamical streaming field (originating from our basic assumption). As the hydrodynamical quantities are defined but as local averages taken over rather big volumes in position space, we can transform the complicated hydrodynamical equations into a much simpler wave equation whose wave function provides the same local averages. There is no modification of the classical equations of motion involved in this procedure. It merely implies a statement about residual velocities (pressure function) and force fluctuations. The wave-mechanical picture implicitly satisfies the minimum relation. In the wave picture we replace our basic hypothesis by the assumption that the distribution (numbers and intensities) of excited ${ensuremath{psi}}_{mathrm{nlm}}$ states $n$ goes up to about ${10}^{6}$) corresponds statistically to the distribution (in numbers and masses) of statistically independent elements of a stellar system. From this and further analysis it is clear that when confronting an observed stellar system with its $ensuremath{psi}$ picture, we have to take averages over volumes containing enough elements to make the hydrodynamical quantities determinate and therefore make inferences about superpositions of many stationary $ensuremath{psi}$ states. None of the discrete effects of ordinary quantum mechanics shows up in the present theory because of this." @default.
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• W2085503627 date "1946-10-01" @default.
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• W2085503627 title "Statistical Hypotheses in Stellar Dynamics" @default.
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• W2085503627 doi "https://doi.org/10.1103/physrev.70.538" @default.
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