Thermodynamic Principles of Stochastic Dynamics: Time Symmetries and Data Infinitum

Abstract

Dynamical systems theory has played a central role in applied mathematics for nearly a century. Besides providing a geometric understanding for difference and differential equations, it is also the main framework for modeling complex systems including those in biology. Complex systems consist of heterogeneous building blocks that are themselves complex objects. Stochastic dynamical models are thus used to represent, statistically, biochemical species in a cell, cells in a tissue, and organisms in an ecosystem. This common stochastic formalism is the foundation for unifying mathematical models in biology at vastly different levels, from cellular biology to terrestrial ecology. We introduce thermodynamic theories from the common stochastic formalism with the following key results: time symmetries dictate dynamic principles and limit theorems in the data infinitum reveal the driving forces conjugated to observables. Our thermodynamic theory is to Markov dissipative dynamics what mechanics is to Hamiltonian dynamical systems. It arises solely from time symmetries and limit theorems in data infinitum and can thus be applied to a wide spectrum of stochastic models in biology and for general complex systems as well. Models in mathematical biology can be integrated in a single universal thermodynamic theory of life.