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 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.

Emergence and Breaking of Duality Symmetry in Generalized Fundamental Thermodynamic Relations

“…The present result augments the current understanding of the nature of thermodynamic behavior, which so far has been focused on large systems limit and phase transition through symmetry breaking as a key route for emergent phenomena [21,22]. We now see there is actually a large measurements limit that generates a different kind of emergent order, a duality symmetry, for any small stochastic systems.…”

Potentials of continuous Markov processes and random perturbations

“…This provides us an orthogonal decomposition of the vector field b = −D∇ϕ + γ where γ⊥∇ϕ as shown by Freidlin and Wentzell (FW) [14]. It is further shown in reference [15] that the decomposition is directly related to the total entropy production decomposition in the small-noise limit. In this paper, we show the general validity of a gradient form of γ, γ = −Q∇ϕ.…”

“…In general, the mesoscopic potential Φ is not a Lyapunov function of the underlying deterministic dynamics in a nonequilibrium diffusion. However, in the small-noise limit, the locallysmooth macroscopic potential ϕ emerges from the globally-smooth Φ and is guaranteed to be the Lyapunov function of the deterministic dynamics x (t) = b(x) [14,15]. This provides us an orthogonal decomposition of the vector field b = −D∇ϕ + γ where γ⊥∇ϕ as shown by Freidlin and Wentzell (FW) [14].…”

Finding meaningful and workable applied mathematics problems in science

“…My Ph.D. group has a tradition of working on thermodynamics of stochastic processes [13,38,17,18,37,4,5]. I worked on a problem of comparing the thermodynamic properties of a Markov chain and its covering space.…”