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A Theory of Mesoscopic Phenomena: Time Scales, Emergent Unpredictability, Symmetry Breaking and Dynamics Across Different Levels | | Ping Ao
; Hong Qian
; Yuhai Tu
; Jin Wang
; | | Date: |
21 Oct 2013 | | Abstract: | 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. | | Source: | arXiv, 1310.5585 | | Services: | Forum | Review | PDF | Favorites | |
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