A Theory of Mesoscopic Phenomena: Time Scales, Emergent Unpredictability, Symmetry Breaking and Dynamics Across Different Levels

Integrating symmetry breaking originally advanced by Anderson, bifurcation from nonlinear dynamical systems, Landau's phenomenological theory of phase transition, and the mechanism of emergent rare events first studied by Kramers, we propose a mathematical representation for mesoscopic dynamics which links fast motions below (microscopic), movements within each discrete state (intra-basin-of-attraction) at the middle, and slow inter-attractor transitions with rates exponentially dependent upon the size of the system. The theory represents the fast dynamics by a stochastic process and the mid-level by a nonlinear dynamics. Multiple attractors arise as emergent properties. The interplay between the stochastic element and nonlinearity, the essence of Kramers' theory, leads to successive jump-like transitions among different basins. We describe each transition as a dynamic symmetry breaking exhibiting Thom-Zeeman catastrophe and phase transition with the breakdown of ergodicity (differentiation). The dynamics of a nonlinear mesoscopic system is not deterministic, rather it is a discrete stochastic jump process. Both the Markov transitions and the very discrete states are emergent phenomena. This emergent inter-attractor stochastic dynamics then serves as the stochastic element for the nonlinear dynamics of a level higher (aggregates) on an even larger spatial and longer time scales (evolution). The mathematical theory captures the hierarchical structure outlined by Anderson and articulates two types of limit of a mesoscopic dynamics: A long-time ensemble thermodynamics in terms of time t, followed by the size of the system N, tending infinity, and a short-time trajectory steady state with N tending infinity followed by t tending infinity. With these limits, symmetry breaking and cusp catastrophe are two perspectives of a same mesoscopic system on different time scales.