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