For recurrent dynamical systems with an invariant measure and repeated measurements {\it ad infinitum}, an entropy function and it’s maximization are emergent phenomena.  In the theory of probability, this is known as Sanov’s  large deviation theory and its contraction.  Convex optimization leads to a thermodynamic like manifold and the concept of “ensembles”.  Legendre transform follows Lagrange duality; Fenchel-Young inequality provides a brand new notion of “equilibrium”.    We argue the concept of energy is the “information” hidden in counting frequency data; and a statistical analysis naturally gives rise to a macro-micro dichotomy.

Professor Hong Qian is Olga Jung Wan Endowed Professor of Applied Mathematics at University of Washington, Seattle. He received his B.A. in Astrophysics from Peking University and Ph.D. in Biochemistry from Washington University in St. Louis, and worked as postdoctoral researcher at University of Oregon and Caltech on biophysical chemistry and mathematical biology. He was elected a fellow of the American Physical Society in 2010. Professor Qian's current research interest is the probabilistic foundation of statistical equilibrium and nonequilibrium thermodynamics and their applications in biology.  His recent, coauthored book “Stochastic Chemical Reaction Systems in Biology” was just published by Springer.