Explore the foundations of statistical mechanics in this 56-minute lecture from the USC Probability and Statistics Seminar. Delve into a novel characterization of the Boltzmann distribution, examining its unique properties in describing the distribution of states in systems with a given temperature. Investigate the theorem's core, which relates to symmetries of the convolution semigroup of finitely supported probability measures on natural numbers and symmetries of the multiplicative semigroup of polynomials with non-negative coefficients. Gain insights into the independence of uncoupled systems and how this property uniquely defines the Boltzmann distribution. Learn about the collaborative research efforts behind this exploration, shedding new light on a fundamental concept in statistical physics.