Stochastic Processes and Statistical Mechanics

Statistical thermodynamics delivers the probability distribution of the equilibrium state of matter through the constrained maximization of a special functional, entropy. Its elegance and enormous success have led to numerous attempts to decipher its language and make it available to problems outside physics, but a formal generalization has remained elusive. Here we show how the formalism of thermodynamics can be applied to any stochastic process. We view a stochastic process as a random walk on the event space of a random variable that produces a feasible distribution of states. The set of feasible distributions obeys thermodynamics: the most probable distribution is the canonical distribution, it maximizes the functionals of statistical mechanics, and its parameters satisfy the same Legendre relationships. Thus the formalism of thermodynamics -- no new functionals beyond those already encountered in statistical physics -- is shown to be a stochastic calculus, a universal language of probability distributions and stochastic processes.