Volume 131, Number 5, September 2020
|Number of page(s)||6|
|Published online||21 September 2020|
Kinematic basis of emergent energetics of complex dynamics
Department of Applied Mathematics, University of Washington - Seattle, WA 98195-3925, USA
Received: 23 May 2020
Accepted: 11 August 2020
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.
PACS: 02.50.Ga – Markov processes / 05.70.Ln – Nonequilibrium and irreversible thermodynamics / 05.40.Ca – Noise
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