  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 $\nabla\varphi$ and its
orthogonal field $\gamma(x)\perp\nabla\varphi$, a general vector field $b(x)$
can be decomposed into $-D(x)\nabla\varphi+\gamma$, where
$\nabla\cdot\big(\omega(x)\gamma(x)\big)=$ $-\nabla\omega D(x)\nabla\varphi$.
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 $d\varphi(x(t))/dt=\gamma D^{-1}\gamma-bD^{-1}b$,
reflecting the geometrical $\|D\nabla\varphi\|^2+\|\gamma\|^2=\|b\|^2$. 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
$\epsilon\to 0$. The present theory provides a mathematical basis for P. W.
Anderson's emergent behavior in the hierarchical structure of complexity
science.
