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Particle number fluctuations and their relation to geometrical probability Ehlers, Peter Frank

Abstract

Fluctuations in the number of particles N within a small subvolume of a system of classical non-interacting particles undergoing free flight, diffusion or Brownian motion are studied. Emphasis is on the temporal correlations and the subvolume geometry. Smoluchowski's method of probability after-effects is used in conjunction with an "intersection volume" technique. Free-flight autocorrelation functions for N(t) are derived for several velocity distributions, including the classical and relativistic Maxwellian. Low-frequency number spectra are obtained for the classical ideal gas in thermal equilibrium when the subvolume is a thin slab, long cylinder or sphere. A new theorem in geometrical probability is obtained by relating the autocorrelations of N and N. We also generalize the problem of the first passage time (to the surface) of particles diffusing out of an n-sphere under absorptive boundary conditions. We account for particle decay and random initial position. The resulting characteristic function of the first passage time is applied to the problem of positronium diffusion in solid powders. The experimentally observed dependence of positron annihilation spectra on powder particle size permits one to calculate the diffusion constant for o-Ps atoms in the powder particles. Diffusion constants deduced by Brandt and Paulin (1968) from their experiments are shown to be too large by a factor of about 4. Corrected values have now been published (Brandt and Paulin, 1972) as a result of discussion with the present writer.

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