**34th Meeting of the AAS Division on Dynamical Astronomy, May 2003**

* 6 Poster Papers*

Posters, Monday, May 5, 2003, 8:00pm,
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## [6.17] Cutoff Energy Behavior for an Ideal Gas

*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 =c_{1}( 1-x)
^{\left( 3N-5\right) /2}\sqrt{x}dx,\quad c_{1}= {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 =c_{2}\rho \sqrt{y}\,dy,\quad c_{2}=(
\int_{0}^{y_{c}}\rho \sqrt{y}\, dy) ^{-1},\quad
y=y_{avg}N{\varepsilon /E} where y_{c}\ 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 y_{c}={\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
\varepsilon_{2c}~{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.