Derivation of the thermodynamic probabilities using polynomial forms

Background and Objectives: A. Einstein suggested at the time that neither the founder of the statistical mechanics L. Boltzmann, nor M. Planck, who has developed the statistical understanding of the entropy of radiation, gave the proper definitions of the thermodynamic probability as a number of equally probable microstates. Meanwhile it was necessary to understand the difference between the Boltzmann’s and Planck’s numbers of microstates. In particular, the Planck thermodynamic probability is expressed by the formula for the sum of the Boltzmann thermodynamic probabilities. The state of the light particles and the state of the Fermi gas have their own peculiarities. In this regard, such approaches are desirable, in which microstates and macroscopic states of different systems can be considered and compared in a single way. Methods: In this paper, the polynomial probability distribution is used to distinguish between distinct definitions of the thermodynamic probability. In the equation that defines the polynomial form, the arguments of the polynomial are supposed to be equal to the probabilities characterizing a micro particle or an oscillator or a phase cell. The degree of polynomials is equal to the numbers of such elements. Each term in the decomposition of the polynomial in powers of the arguments gives the probability of a state in which the individual elements of the system are indistinguishable, that is, the probability of the observed macroscopic state. Using this scheme, it is sometimes possible to trace the correlation between the probabilities of the microstates of the individual elements and the probabilities of the macroscopic states of the system. Conclusion: The Boltzmann thermodynamic probabilities for the classical molecular gas, the Planck thermodynamic probability for emitting quantized oscillators, the Bose thermodynamic probability for photons of thermal radiation, and the thermodynamic probability for Fermi gas are considered in the present paper within the framework of the proposed approach. It is traced how the differences in thermodynamic probabilities are shown in polynomial forms. In particular, the article shows the refinement of the statement that the sum of the Boltzmann thermodynamic probabilities is the same, as the Planck thermodynamic probability.