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M. Cahill (U. Wisconsin, Washington County)
The energy distribution for an ideal gas is important to dynamical astronomy because it is used as the statistical basis for modeling relaxed dynamical systems. This presentation deals with some fundamental aspects of this distribution.
The microcanonical distribution for a monatomic ideal gas gives the probability that a particle's energy is in a specified range simply as d\psi =c1( 1-x) \left( 3N-5\right) /2\sqrt{x}dx,\quad c1= {2\over \sqrt{\pi }}{\Gamma ({3N/2}) \over \Gamma ({3[ N-1 ] 2})},\quad x={\varepsilon /E}, where N and E are the total population and total energy of the gas while \varepsilon \ and x are the kinetic energy of the particle in usual and dimensionless forms. The cutoff energy for this distribution occurs when \varepsilon =E.
An alternative distribution for this gas may be obtained through most probable methods but it is more complex and given by d\psi =c2\rho \sqrt{y}\,dy,\quad c2=( \int0y_{c}\rho \sqrt{y}\, dy) -1,\quad y=yavgN{\varepsilon /E} where yc\ is the cutoff value of the dimensionless energy y and \rho \ is the velocity space density of the phase points of the gas, which can be shown to be given by D( \rho ) ={\ln ( N!) / N}-y,\quad D( z) ={d\ln ( z!) / dz},\quad yc={\ln ( N!) / N}+\gamma , where \gamma is Euler's constant.
For the large N case, the energy cutoffs for the microcanonical distribution, \varepsilon 1c, and that for the most probable distribution, \varepsilon 2c, are very different and approximately obey \varepsilon2c~{E\ln N / N}={\varepsilon 1c\ln N / N}. This apparent difficulty is resolved by numerical studies showing that the number of particles in the mirocanonical distribution with energies above \varepsilon 2c is negligibly small in comparison to N.
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Bulletin of the American Astronomical Society, 35 #4
© 2003. The American Astronomical Soceity.