A framework towards understanding mesoscopic phenomena: Emergent unpredictability, symmetry breaking and dynamics across scales

Abstract

By integrating 4 lines of thoughts: symmetry breaking originally advanced by Anderson, bifurcation from nonlinear dynamics, Landau's theory of phase transition, and the mechanism of emergent rare events studied by Kramers, we introduce a possible framework for understanding mesoscopic dynamics that links (i) fast lower level microscopic motions, (ii) movements within each basin at the mid-level, and (iii) higher-level rare transitions between neighboring basins, which have rates that decrease exponentially with the size of the system. In this mesoscopic framework, multiple attractors arise as emergent properties of the nonlinear systems. The interplay between the stochasticity and nonlinearity leads to successive jump-like transitions among different basins. We argue each transition is a dynamic symmetry breaking, with the potential of exhibiting Thom-Zeeman catastrophe as well as phase transition with the breakdown of ergodicity (e.g., cell differentiation). The slow-time dynamics of the nonlinear mesoscopic system is not deterministic, rather it is a discrete stochastic jump process. The existence of these discrete states and the Markov transitions among them are both emergent phenomena. This emergent stochastic jump dynamics then serves as the stochastic element for the nonlinear dynamics of a higher level aggregates on an even larger spatial and slower time scales (e.g., evolution). This description captures the hierarchical structure outlined by Anderson and illustrates two distinct types of limit of a mesoscopic dynamics: A long-time ensemble thermodynamics in terms of time $t$ tending infinity 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 the same mesoscopic system on different time scales.

30 pages, 3 figures, 1 table

References in corpus (6)

• Stochastic thermodynamics, fluctuation theorems, and molecular machines
• Stochastic Dynamical Structure (SDS) of Nonequilibrium Processes in the Absence of Detailed Balance. IV: Emerging of Stochastic Dynamical Equalities and Steady State Thermodynamics from Darwinian Dynamics
• Mesoscopic Kinetic Basis of Macroscopic Chemical Thermodynamics: A Mathematical Theory
• Curl flux, coherence, and population landscape of molecular systems: Nonequilibrium quantum steady state, energy (charge) transport, and thermodynamics
• Analytical Mechanics in Stochastic Dynamics: Most Probable Path, Large-Deviation Rate Function and Hamilton-Jacobi Equation
• Nonlinear Stochastic Dynamics of Complex Systems, I: A Chemical Reaction Kinetic Perspective with Mesoscopic Nonequilibrium Thermodynamics

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• Dynamical mean-field theory: from ecosystems to reaction networks
• Ergodicity Breaking and Scaling Relations for Finite-Time First-Order Phase Transition
• Representations and Divergences in the Space of Probability Measures and Stochastic Thermodynamics
• Universal Ideal Behavior and Macroscopic Work Relation of Linear Irreversible Stochastic Thermodynamics
• Noise equals control
• Dynamics of systems with varying number of particles: from Liouville equations to general master equations for open systems
• From Near-Integrable to Far-from-Integrable: A Unified Picture of Thermalization and Heat Transport
• Mesoscopic and Macroscopic Entropy Balance Equations in a Stochastic Dynamics and Its Deterministic Limit