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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.

This tutorial presents a mathematical theory that relates the probability of sample frequencies, of M phenotypes in an isogenic population of N cells, to the probability distribution of the sample mean of a quantitative biomarker, when the N is very large. An analogue to the statistical mechanics of canonical ensemble is discussed.

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