Statistical Considerations of the Dirac Type 2x2 Linear Matrix Equation

The 2x2 Dirac-like matrix linearization of the energy-momentum equation can be interpreted as a probability scenario. It was argued in earlier notes that this was in fact a quantum mechanical probabilistic approach as it leads to the Schrodinger zitterbewegung . We have also seen that the probabilities .5(1+v) and .5(1-v) associated with the energy eigenvector .5( sqrt(1+v), sqrt(1-v)) can also be obtained by finding a stationary solution of the entropy given by  -k f1 ln(f1) -k f2 ln(f2) - w (f1-f2) where the constraint f1-f2=v has been used. The unnormalized results are f1=sqrt(1+v)/sqrt(1-v) and f2=1/f1, i.e. the two Doppler shift scenarios. Kaniadakis has developed a relativistic statistical model in which the Maxwell-Boltzmann distribution only holds in the nonrelativistic case. He instead uses for entropy:

S= 1/2k Integral d3p [f(1-k)/(1-k) - f(1+k)/(1+k)]

One can see that f1 and f2=1/f1 together with the constraint f1-f2 =v are also stationary solutions of this new relativistic entropy. In fact, we suggest that one could use the f1 and f2=1/f1 result to postulate Kaniadakis entropy expression such that it becomes the natural logarithm in the k->0 (nonrelativistic) limit. Kaniadakis’ new expression for the natural logarithm is the sum of the mathematical limits    -ln(a)=lim x->0 [a-x -1]/x   and ln(a) = lim x->0  [ax -1]/x. F1 and f2 from the 2x2 matrix model should already be relativistic expressions and so are not “nonrelativistic limits”. It turns out they provide stationary solutions to both Kaniadakis’ relativistic entropy expression and the normal  S= -k f ln(f) expression which leads to the Maxwell-Boltzmann result. The experimental motivation for a new relativistic distribution is that relativistic data apparently does not match the tail the Maxwell-Boltzmann distribution.