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 $\phi (x)$ emerges as the limit of the generalized, nonequilibrium free energy of a Markovian dynamics with vanishing fluctuations. In terms of the $\nabla \phi$ and its orthogonal field $\gamma (x)\perp \nabla \phi$, a general vector field $b(x)$ can be decomposed into $-D(x)\nabla \phi +\gamma$, where $\nabla \cdot (\omega (x)\gamma (x))=$ $-\nabla \omega D(x)\nabla \phi$. 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 $x$. $\phi (x)$ and $\omega (x)$ are interpreted as the emergent energy and degeneracy of the motion, with an energy balance equation $d\phi (x(t))/dt=\gamma D^{-1}\gamma -b{D}^{-1}b$, reflecting the geometrical $\Vert D\nabla \phi {\Vert }^2+\Vert \gamma {\Vert }^2=\Vert b{\Vert }^2$. 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 $\epsilon \to 0$. The present theory provides a mathematical basis for P. W. Anderson's emergent behavior in the hierarchical structure of complexity science.

7 pages

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