Reduced dynamics and geometric optimal control of nonequilibrium thermodynamics: Gaussian case

Abstract

This paper introduces a control-theoretic formulation of nonequilibrium thermodynamic systems with Gibbs states of Gaussian distributions, i.e., thermodynamic systems characterized by Gaussian probability densities. The distinct features of the paper are three-fold. First, the dynamics of a thermodynamic system is studied by transferring the original state variable from the non-Euclidean and nonlinear manifold state space to a Lie group, and further to the dual space of a Lie algebra, which is endowed with vector space structures. Consequently, the resulting reduced dynamics significantly reduces the high nonlinearity appearing in the original non-Euclidean state space of a thermal process. Second, the obtained equations of motion interestingly indicate that thermodynamics can be naturally viewed as a generalization of rigid body motions, and this bridges control theory, thermodynamics, information theory, and rigid body dynamics. Third, the optimality conditions for the energy-minimum optimal control problem of probability densities are derived via geometric Pontryagin’s principle by regarding the reduced dynamics as dynamical constraints, and a unified control algorithm is developed. Finally, the proposed approach is applied to three different scenarios including two benchmark examples to demonstrate the applicability and effectiveness. The purpose of the paper is to provide a deeper understanding of thermodynamics from a control perspective, and also to draw intrinsic connections between control theory, thermodynamics, information theory, and rigid body dynamics.