{
  "id": "1310.5585",
  "version": "v2",
  "published": "2013-10-21T15:04:12.000Z",
  "updated": "2016-11-03T02:05:59.000Z",
  "title": "A framework towards understanding mesoscopic phenomena: Emergent unpredictability, symmetry breaking and dynamics across scales",
  "authors": [
    "Hong Qian",
    "Ping Ao",
    "Yuhai Tu",
    "Jin Wang"
  ],
  "comment": "30 pages, 3 figures, 1 table",
  "journal": "Chemical Physics Letters, Vol. 665, pp. 153-161 (2016)",
  "doi": "10.1016/j.cplett.2016.10.059",
  "categories": [
    "cond-mat.stat-mech"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-10-21T15:04:12.000Z",
      "title": "A Theory of Mesoscopic Phenomena: Time Scales, Emergent Unpredictability, Symmetry Breaking and Dynamics Across Different Levels",
      "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.",
      "comment": "23 pages, 3 figures, 1 table",
      "journal": null,
      "doi": null,
      "authors": [
        "Ping Ao",
        "Hong Qian",
        "Yuhai Tu",
        "Jin Wang"
      ]
    },
    {
      "version": "v2",
      "updated": "2016-11-03T02:05:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "time scales",
      "emergent unpredictability",
      "mesoscopic phenomena",
      "transition",
      "breaking exhibiting thom-zeeman catastrophe"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1310.5585A"
    }
  }
}