Speculation on Maxwell-Boltzmann Distribution From a Microcanonical Power Law

 In (1), a power law distribution is obtained through geometric (using n-sphere and n-1 sphere surface areas) means for n  one-dimensional particles with a total energy E, i..e. the condition of the microcanonical distribution is imposed. In the large n limit, this power law distribution becomes the Maxwell-Boltzmann distribution. The geometrical approach is based on the equation:  p1p1 + p2p2 + … +pnpn= RR where E=RR/2m (nonrelativistic case), where each pi is like a co-ordinate.

    We try to argue that the result of (1), namely f(p) = (1-pp/RR) power( (n-3/)/2)  may be obtained without using the geometric n dimensional sphere approach of (1). We write: e1+e2+..en = E and argue that ultimately each e should have the same probability and degeneracy. The degeneracy arises from:  de= d (pp/2m) = p dp/2m or C1 sqrt(e) dp.  In (1), a distribution as a function of p is desired. For a given e and E, the energy which may be distributed is E-e. We argue that for n=3, the probability should be the same for all particles. We further suggest a product of degeneracy values and thus suggest:   sqrt(E-e) power(n-3). This is the exact result of (1). In the case of 3-dimensional particles, we apply the degeneracy also to dpx dpy dpz so sqrt(e) becomes sqrt(e)sqrt(e)sqrt(e). This then yields the correct relationship between energy and T (temperature) in the Maxwell-Botlzmann large n limit.