Tsallis and Kaniadakis Distributions Derived Without Explicit Use of Conservation of Energy

In (1), it is stated that people are concerned about the use of Tsallis and Kaniadakis distributions due to the “ existence of a limit of validity for ordinary statistical mechanics and the reason for its occurrence.” (1) In other words, there is a question as to why the Tsallis and Kaniadakis distributions should exist both theoretically and in nature instead of simply the Maxwell-Boltzmann distribution, it seems. We have developed a stability procedure (2), which produces a single differential equation: F(p(ei)) + p(ei) dF/dp(ei) = a1ei + a2 for the case of constraints Sum over i p(ei)=1 and Sum over i ei p(ei) = Eave. Here F(p(ei)) = -ei/T (i.e. the constraint ei form information function) and p(ei) is unnormalized. The Maxwell-Boltzmann, Tsallis and Kaniadakis distributions are solutions of this single differential equation, which is linked to physical stability under p(ei)->p(ei)+dp(ei). As a result, all three distributions are physically stable. Here we wish to consider why the Tsallis and Kaniadakis distributions should exist from a different point of view. We suggest that the Maxwell-Boltzmann distribution derivation is very closely linked to conservation of energy. In fact, it is the explicit link of this statistical distribution to conservation of energy which yields its form p(ei) = C(T) exp(-ei/T) because p(ei)p(ej)=exp(- (ei+ej)/T). We suggest that even though conservation of energy exists, there might be other relationships, i.e. p(ei)p(ej)=p(ek)p(el) for say ei*ej = ek*el. In this equation ei+ej not= ek+el, but that does not mean that one violates conservation of energy. It just means that one cannot describe the probability for two particles to interact as p(ei)p(ej) which is fine physically. Certainly p(ei)p(ej) is not the full probability description in the Bose-Einstein and Fermi-Dirac cases and so it does not need to be the full description in other cases as well. Even if ei and ej do not react to produce ek,el, (due to conservation of energy imbalance), this does not a priori mean that ther might not be a relationship p(ei)p(ej)=p(ek)p(el) (even for ei+ej not=ek+el). We argue that one has to be careful about forcing conservation of energy to define the statistical distribution and then arguing that only one distribution can essentially exist in nature, namely the Maxwell-Boltzmann one. ( Distributions such as Gaussians, the Poisson and uniform are linked to different scenarios.)