A Generalized Form of the New Statistics

Instead of assuming the probabilities of absorption and emission of atoms in a radiation field as equal to B_aρ and A+B_eρ (the well-known Einstein equations) Strum has in a recent paper assumed them to be respectively equal to B_aρ+C_aρ²+⋯ and A+B_eρ+C_eρ²+⋯. This leads to a modification of Planck's law. It is shown in this paper that this assumption effectively amounts to replacing the new statistical law of interaction of Nordheim e.g. f(1±f^*/A)f₁^*(1±f₁^*/A₁) by the more general (f+af²+⋯)(1±f^*/A±af^*2/A±⋯)(f₁+af₁²+⋯)(1±f₁^*/A₁±af₁^*2/A₁±⋯) where f, f₁, f^*, f₁^* are the distribution functions for the projectiles corresponding to the numbers of each before and after collision; A, A₁ the number of cells respectively associated with them. On this basis a general H function defined by H=∫(f+af²+⋯) log(f+af²+⋯)∓(A±f±af²±⋯)log(A±f±af²±⋯) is considered. As a consequence of this we get a generalized distribution function given by the relation f+af²+bf³+⋯=A/Cexp[c|v|²]∓1 If we take only two terms we get approximately J=A/Ce^c|v|2∓1-A²a/(Ce^c|v|2∓1)²=A/Cexp[c|v|²]∓1-A²a/(Cexp[c|v|²]∓1)². By substituting the appropriate values for A in the cases corresponding to the photons and the atoms, we get Strum's modified radiation formula, and a new distribution function for a gas. Thus as a result of a generalized analysis of the H-theorem we get a new statistics the successive approximations of which are a more accurate statistics which includes Strum's formula, the statistics of Bose-Einstein, Fermi-Dirac, the classical statistics of Maxwell-Boltzmann. It is suggested that this new statistics may be of importance in the study of dense matter at high temperature.