Tsallis/Kaniadakis Distributions, Conservation of Energy and Probability Balance

The Maxwell-Boltzmann distribution is interesting in that it is associated with a gas with fixed energy, but each elastic collision is linked to probability through p(ei)p(ej)=p(ek)p(el) such that ei+ej=ek+el. This means that if one considers time reversal balance, each type to two body collision conserves p(ei) values. Another way to state this is that conservation of probability in a single type of 2-body collision is directly linked to conservation of energy. In (1), we introduced a stability method which yields the Maxwell-Boltzmann, Tsallis and Kaniadakis distributions through a single approach (differential equation) based on the ei constraint. This ei constraint (the first moment in ei) is Sum over i eip(ei) =Eave suggesting conservatin of energy. Yet for the Tsallis and Kaniadakis cases, one has ei*ej=ek*el for e/T>>1. As a result, for e/T>>1 a given elastic collision ei+ej = ek+el does not conserve probability (considering time reversal), unlike the MB case. The distribution itself for e/T>>1, namely a power law drop (ei/T) power -b must ensure that overall probability is conserved. We try to show that this is the case in the following manner. For ei*ej, we set ei=b and ej=b+d. An ek,el set such that ei+ej=ek+el means that: ek= b+d1 and el =b+d-d1. The product ei*ej differs from ek*el by the term d(d-d1). This means that if ei*ej>ek*el then due to the power law, ek*el produces more ei,ej pairs than ei*ej produces ek,el pairs and there is an imbalance. One may, however, consider d,d1>0 with d1>d and d1<d to see that these two scenarios balance each other . Similarly, one may consider d1,d<1 which produces the same result as expected. d1>0, d<0 is a little different, but there is a balance for d1>d and d1<d. As a rresul, it seems that ei*ej=ek*el and its corresponding power law does allow for conservation of probability even if collisions are elastic.