Distribution and entropy of Boltzmann as infinite convergent consequences

Abstract

The equilibrium Boltzmann distribution is an important and strict tool for the definition of entropy, since this function is not measured and only calculated in accordance with the Boltzmann law. On the basis of the coefficient of proportionality of discrete and continuous similar distributions developed by the authors, an analysis is made of the partition function in the Boltzmann distribution to the commensurability with the improper integral of the function of the same name in the full range of the terms of the partition function for different combinations of temperature and the step of varying the particle energy. The convergence of the series based on the Cauchy and Maclaurin criterion and the equal proportionality of the series and the improper integral of
the function of the same name in each unit interval of variation of the series and the function of the same name are established. The obtained formulas for the coefficient of proportionality and the partition function are analyzed, and a general expression is found for the total and residual statistical sums, which can be calculated with any given accuracy. Given a direct calculation formula for the Boltzmann distribution, taking into account the values of the improper integral and the coefficient of proportionality. To determine the entropy from the new expression for the Boltzmann distribution in the form of a series, the convergence of the improper integral with the same name is established. However, the coefficient of proportionality of the integral and the «entropy» series in each unit interval turns out to be dependent on the number of the term of the series and therefore can not be used to determine the sum of the series in terms of an improper integral. In this case, the calculation of the entropy can be carried out with a specified accuracy with a corresponding number of terms of the series n for a fixed value of the partition function, and at high temperatures – by direct calculation through the coefficient of proportionality and the improper intergal. The given accuracy of the statistical sum turns out to be mathematically identical to the fraction of particles with an energy exceeding a given energy barrier level equal to the activation energy in the Arrhenius equation. The prospect of the development of the proposed method for expressing Boltzmann's distribution and entropy is to establish the relationship between the quantum of energy ∆ε and the properties of system-forming particles, and also taking into account the information degeneracy of the thermodynamic system at an infinitely high temperature.