Gamma Function Versus Maxwell-Boltzmann Distribution

   In (1), a gamma function (e/T) power (k-1) exp(-e/T) is proposed instead of the usual Maxwell-Boltzmann distribution exp(-e/T). It is argued that there is not enough experimental data which verifies the MB distribution and so the gamma function may be studied.

   Here we consider physical implications of the gamma function. We first note that a main idea of the MB distribution is that exp(-e1/T)exp(-e2/T) = exp(-e3/T)exp(-e4/T) for an elastic collision e1+e2=e3+e4. We note that gamma function retains this property, but only for energies near kT. We also note that given the integral de sqrt(e) exp(-e/T), introducing an (e/T) power (k-1) (as in (1)) does not change the form of the rising integral from e/T=0. It does, however, change the fall of the distribution, making it less steep as we show. Finally, we consider obtaining the gamma function distribution from an extremization of entropy and show that one no longer uses total energy sum over i ei p(ei)=Etotal as the constraint (which is a physical a priori constraint), but rather sum over i p(ei) { (k-1) ln(ei/T) - ei/T) } = number. We show that one may also consider reaction balance by replacing p(ei) with p(ei) / (ei/T) power (k-1). In other words, two body scattering is more complicated under the gamma function scenario except for ei/T values near 1.