A Foundation of the Boltzmann Distribution, or Tiltings as the Unique Support-Preserving Convolution Endomorphisms

Abstract: The family of Boltzmann distributions is used in statistical mechanics to describe the distribution of states in systems with a given temperature. We give a novel characterization of this family as the unique one satisfying independence for uncoupled systems. The theorem boils down to a statement about endomorphisms of the convolution semigroup of finitely supported probability measures on the natural numbers, or, alternatively, about endomorphisms of the multiplicative semigroup of polynomials with non-negative coefficients.