Affine Structures and Stochastic Thermodynamics on the Space of Measures

Abstract

Kolmogorov’s theory of probability emphasizes a given state space Ω and a given probabilitymeasure P, then constructs the entire calculus of measurable functions X : Ω → R. From this perspective, the properties and dynamics of given families of probability measures are viewed as technical subjects within the general theory. In this work, we show that two fundamental concepts from statistical physics - entropy and energy - are themselves stochastic objects when one considers change of measure to be the natural representation of real-world dynamics. A relationship between thermodynamics and probability theory is formulated in terms of large deviation principles and affine structures on the space of measures.