Stochastic kinematic description of a complex dynamics is shown to dictate an
energetic and thermodynamic structure. An energy function Ο(x) emerges
as the limit of the generalized, nonequilibrium free energy of a Markovian
dynamics with vanishing fluctuations. In terms of the βΟ and its
orthogonal field Ξ³(x)β₯βΟ, a general vector field b(x)
can be decomposed into βD(x)βΟ+Ξ³, where
ββ (Ο(x)Ξ³(x))=ββΟD(x)βΟ.
The matrix D(x) and scalar Ο(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. Ο(x) and Ο(x)
are interpreted as the emergent energy and degeneracy of the motion, with an
energy balance equation dΟ(x(t))/dt=Ξ³Dβ1Ξ³βbDβ1b,
reflecting the geometrical β₯DβΟβ₯2+β₯Ξ³β₯2=β₯bβ₯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
Ο΅β0. The present theory provides a mathematical basis for P. W.
Anderson's emergent behavior in the hierarchical structure of complexity
science.Comment: 7 page