Kinematic basis of emergent energetics of complex dynamics

Abstract

Stochastic kinematic description of a complex dynamics is shown to dictate an energetic and thermodynamic structure. An energy function emerges as the limit of the generalized, nonequilibrium free energy of a Markovian dynamics with vanishing fluctuations. In terms of the ∇φ and its orthogonal field , a general vector field can be decomposed into , where . The matrix and scalar , two additional characteristics to the alone, represent the local geometry and density of states intrinsic to the statistical motion in the state space at . and are interpreted as the emergent energy and degeneracy of the motion, with an energy balance equation , reflecting the geometrical . The partition function employed in statistical mechanics and Gibbs' method of ensemble change naturally arise; a fluctuation-dissipation theorem is established via the two leading-order asymptotics of entropy production as . The present theory provides a mathematical basis for Anderson's emergent behavior in the hierarchical structure of complexity science.