From Statistical Mechanics to Process Control: A Tutorial

Abstract

A system-theoretic basis for process control is developed using statistical mechanics as our point of departure. Specifically we show that control of a thermodynamic system can be discussed within the framework of passive systems theory. Using Boltzmann's definition of the entropy S = kB ln(Ω) and assuming that the distribution of states is uniformly distributed close to the equilibrium manifold we demonstrate the existence of a Lyapunov function based on the concept of available work. The LaSalle invariance principle now applies and we conclude that the microscopic (quantum) state converges to an ω-limit set close to the equilibrium manifold. The support of the measure V, interpreted as the volume of the system, defines the phase distribution in a multi-phase system. To connect such ideas with process control we define (approximate) balance equations, force, and flux variables.