Energy shifts and Kaniadakis Entropy

Recently, Kaniadakis proposed a new statistical distribution for relativistic particles. His expression becomes the Maxwell-Boltzmann (MB) distribution under nonrelativistic conditions. It is interesting to note, however, that the MB distribution is not affected by a constant shift in the value of energy. If energy goes from E to E+b the (MB) distribution exp(-E/T)/C, where C is a normalization constant, stays unchanged. It is further interesting to note that the MB distribution can be obtained through a variational calculation of: Sum fi ln(fi)-fi - w( Sum ei fi) with Sum fi = 1 and Sum ei fi =E/N where E is the total energy and N the total number of particles and w a Lagrange multiplier. Now, the energy constraint can be shifted to give a new constraint Sum (ei-b) fi = E/N -b. Using this in a variational calculation would give a distribution proportional to exp( -(ei-b)/T), but this should not lead to any physical changes. In addition, examining the microcanonical distribution seems to indicate that a shift in energy should not change the number of states or the probability distribution. Curiously, the distribution given by Kaniadakis does not appear to be invariant under a shift in energy. The purpose of this note to examine energy shift and to see if there is a simple way to add this invariance to the Kaniadakis distribution.