Physics – Condensed Matter – Statistical Mechanics
Scientific paper
May 2000
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2000dda....31.1503c&link_type=abstract
American Astronomical Society, DDA Meeting #31, #15.03; Bulletin of the American Astronomical Society, Vol. 32, p.871
Physics
Condensed Matter
Statistical Mechanics
Scientific paper
An analysis of the statistical mechanics of spherical star clusters and the simpler case of monatomic ideal gases is known to reveal that the Michie-King globular cluster energy cutoff can be written as \varepsilonc ln ( N ) /line\varepsilon where \varepsilonc is the cutoff energy, /line=\varepsilon is the average energy, the masses are assumed identical and where the number of cluster stars or gas particles, N, is large. The statistical methods leading to this result are shown to work for a photon gas. The usual method of dividing phase space into cells each with Λ quantum compartments to which Bose-Einstein statistics apply is used. The resulting distribution law is D (Λ -1+Λ ρ ) -D ( Λ ρ ) =\varepsilon/ \varepsilono where D is the logarithmic derivative of the factorial function, \varepsilon and \varepsilon o are respectively the energy variable and an energy constant characterising the distribution and where ρ is the quantum compartment particle number density ρ =h3dn/dω , with\ dω being the phase space volume element. The distributions ρ ( \varepsilon /\varrepsilono, Λ ) are shown to be a one parameter family of distributions which approach the Planck law as Λ approaches ∞ . For large Λ , the photon density falls to 0, where it is cut off, at \varepsilon = \varepsilonc ln ( Λ ) /line\varrepsilon. If the energy of a photon gas is finite, there must be a frequency cutoff since photon energy is proportional to frequency. It follows that Λ is finite, it is shown that Λ N, and that the Planck law is an excellent approximation for its distribution except in the tail region. Also the last 2 equations imply equation 1 holds for the photon gas.
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