Boltzmann versus Kaniadakis Entropy for Fermi Dirac and Bose Einstein Cases

 In an earlier note (1), it was argued one may derive the Fermi-Dirac (FD), Bose-Einstein (BE) (and Maxwell-Boltzmann (MB)) distributions using only time reversal balanced elastic reactions. For example, given e1+e2 =e3+e4 ((1a)) one may find dynamical factors needed for such a reaction. For the FD case, one may use f(e1)(1-f(e3))f(e2)(1-f(e4)) = f(e2)(1-f(e1)) f(e4)(1-f(e2))  ((1b)). Taking the ln of ((1b)) and equating to energy conservation yields the FD distribution. A similar approach applies to the MB and BE cases. Thus, no information about phase space and no concept of entropy is needed. Following Kaniadakis (2), one may formally write a kinematical function k such that ln(k(f(e1)) + ln(k(f(e2)) = ln(k(f(e3)) + ln(k(f(e4)) ((2)). Thus: ln(k) is the inverse of f(e1). Kaniadakis integrates ln(k(f(e)) over df to obtain an entropy density which may coupled with the Lagrange multiplier b f(e) e  (i.e. an overall fixed energy). The integral and derivative d/df are inverse functions so maximization leads to:

ln(k(f(e)) = -be   ((3)) which follows immediately from a time reversal balance approach i.e.  ((1a)) and ((1b)) with no entropy needed.

      The focus of this note, however, is to compare Kaniadakis entropy, which is directly linked to kinematics but not to phase space, with the Boltzmann entropy approach where entropy S=-k ln(Number of States). In (3), the number of states is computed using various factorials which seem to incorporate phase space. These approaches of course yield the MB, FD and BE distributions, but only after one applies Stirling’s approximation. It may, however, be unnecessary to think in terms of Stirling’s approximation. For example, in information theory, one maximizes a macrostate probability  Pa = Product over i fini  where ni=Nfi, fi being a probability for i, subject to the Lagrange constraint  Sum over i ei fi. This yields the MB distribution, but no factorials appear. Instead, the usual Stirlings result follows from ln(fifi). Thus, it is argued one might interpret the Stirling form of the Boltzmann entropy, which is very similar to the Kandiadkis entropy, in such a manner instead of in terms of counting states. The reason that counting states seems a little strange, at least for the BE case, is that there should be an enhancement in the probability Pn for a state with n bosons (4). This physical enhancement is plainly visible in the time reversal balance approach ((1a)) ((1b)), but does not seem to be clearly present for a counting scheme which gives each “arrangement” an equal weight of 1.