Speculation on Velocity Enhancement of f(e) Distributions and the Maximization of Entropy

 The Maxwell-Boltzmann distribution may be obtained by considering a time reversal balance for elastic 2-body scattering. This approach assumes that:  f(e1)f(e2) =f(e3)f(e4) together with e1+e2=e3+e4. Taking ln of the first equation and equating with -1/T of the second yields the MB distribution f(e)=Cexp(-e/T). This result is equivalent to considering the ln of the number of ways a total energy E may be distributed among N particles, i.e. ln{ N!/ Product over i n(ei)! }. Maximizing subject to the relaxed constraint, Sum over i ei f(ei), also yields the MB distribution.

   We have argued in previous notes that a collision may depend not only on f(e), but on a function of speed which increases with speed. In other words, a fast moving particle moves through a greater distance in delta time and has more of a chance of colliding than one would assume by its f(e), we argue. This function g(e) (based on speed) would be a known function (from experiment) and would not be associated with any kind of variation of f(e). From a 2-body elastic collision viewpoint, it seems that one simply replaces f(ei) with f(ei)g(ei) and obtains fg = C exp(-e/T). Eave, however, is Sum over i ei f(ei) and so one does not have the average values as in the MB case. Furthermore, E-TS is no longer the simple -kT ln(C(T)) result.

   Here, we consider how these results change the notion of the usual MB entropy and also speculateon Tsallis entropy as we have argued in previous notes that q is an attempt to modify the weight of e/T  for e/T not <<1.