Kinematic Basis of Emergent Energetics of Complex Dynamics

Stochastic kinematic description of a complex dynamics is shown to dictate an energetic and thermodynamic structure. An energy function $varphi(x)$ emerges as the limit of the generalized, nonequilibrium free energy of a Markovian dynamics with vanishing fluctuations. In terms of the $ ablavarphi$ and its orthogonal field $gamma(x)perp ablavarphi$, a general vector field $b(x)$ can be decomposed into $-D(x) ablavarphi+gamma$, where $ ablacdotbig(omega(x)gamma(x)big)=$ $- ablaomega D(x) ablavarphi$. The matrix $D(x)$ and scalar $omega(x)$, two additional characteristics to the $b(x)$ alone, represent the local geometry and density of states intrinsic to the statistical motion in the state space at $x$. $varphi(x)$ and $omega(x)$ are interpreted as the emergent energy and degeneracy of the motion, with an energy balance equation $dvarphi(x(t))/dt=gamma D^{-1}gamma-bD^{-1}b$, reflecting the geometrical $|D ablavarphi|^2+|gamma|^2=|b|^2$. The partition function employed in statistical mechanics and J. W. Gibbs method of ensemble change naturally arise; a fluctuation-dissipation theorem is established via the two leading-order asymptotics of entropy production as $epsilonto 0$. The present theory provides a mathematical basis for P. W. Andersons emergent behavior in the hierarchical structure of complexity science.