Arrangements, P(ei)P(ej) = P(ek) and the Maxwell-Boltzmann and Power Law Distributions

  In the statistics of the Maxwell-Boltzmann distribution, one makes use of the idea of elastic collisions and the conservation of energy. Thus, when one maximizes entropy, it is subject to a physical constraint, namely the average energy present. In previous notes, we argued that the main idea behind the MB distribution is:  P(e1)P(e2) = P(e3=e1+e2). In this relation, one sees the explicit notion of energy being distributed i.e. e1+e2= e3. The relation shows that one cannot distinguish between products of probabilities representing the same total energy. This leads to the idea of maximizing the number of arrangements of distributing energy, because all have the same probability.

    In this note, we argue that there seems to be a more general statistical idea at work which is concerned with probabilities more than with the case of energy conservation. In the MB case, the two happen to coincide. In particular, we suggest that one examine: P(e1)P(e2)=P(e3) with e1+e2 not equal e3. One must then find a relationship between e1,e2 and e3. In other words, a given e1 and e2 map to a value e3 such that P(e3) has the same probability as P(e1)P(e2). Given that one may take the logarithm of P(e1)P(e2)=P(e3), this leads to a conservation law as well the form ln(P(ei)) = function(ei) . ln(P) appears in Stirling’s relation:  ln(N!) approx= Nln(N) for N large, so one may consider  d/dP  { P ln(P) } = ln(P) + Constant as being the maximization of the number of arrangements linked with function(ei). This function is only ei/T for the MB case. Thus one maximizes subject to an constraint: Sum over i function(ei) P(ei) which is a mathematical object which accounts for “equal probabilities” i.e. P(e1)P(e2)=P(e3). This does not imply that P ln(P) is entropy because a different recipe is used for this quantity, namely:  dS/dP = a+bei.  In other words, entropy maximization is associated with the constraint Sum over i ei P(ei). We apply these ideas to the power law:  (1-ke/T) (power 1/k) which appears in (1).