Information Theory and Statistical Mechanics Revisited

We derive Bose-Einstein statistics and Fermi-Dirac statistics by Principle of Maximum Entropy applied to two families of entropy functions different from the Boltzmann-Gibbs-Shannon entropy. These entropy functions are identified with special cases of modified Naudts’ φ-entropy. In a pioneering work sixty years ago, Jaynes [4] initiated the study of statistical physics by information theory. He expounded the Principle of Maximum Entropy and applied it to Boltzmann-Gibbs-Shannon entropy to derive Boltzmann-Gibbs statisitics. Many different entropy functionals have been introduced, studied and used in many areas where statistics have been applied. But in physics, only the BoltzmannGibbs-Shanon entropy has been considered as “physical”. Tsallis [7] has proposed the q-entropy as another kind of physical entropy functional. Naudts [6] introduced φ-exponentials, φ-logarithms and φentropy as a further generalization related to the q-entropy. On [6, p. 94] he wrote:“One of the conclusions is that the q-deformed exponential family occurs in a natural way within the context of classical mechanics. The more abstract generalisations discussed in the final chapters may seem less important from a physics point of view. But they have been helpful in elucidating the structure of the theory of generalised exponential families.” We will show that such generalizations are indeed of physical importance, in particular, in understanding the Bose-Einstein statistics and the Fermi-Dirac statistics by Principle of Maximum Entropy. We will first consider the subjective statistical physics of free bosons and free fermions. More precisely, we will derive Bose-Einstein statistics and Fermi-Dirac statistics from the Principle of Maximum Entropy, not for the Boltzmann-Gibbs-Shanon entropy as Jaynes did for the Boltzmann-Gibbs statistics, but instead for two different entropy functions. We quote here Tsallis [7, p. 4]: “Indeed, the physically important entropy a crucial concept is not thought as being an universal functional that is given once for ever, but it rather is a delicate and powerful concept to be carefully constructed for classes of