Non-ergodic transitions in many-body Langevin systems: a method of dynamical system reduction

Department of Applied Life Sciences, The University of Tokyo, Tōkyō, Japan
Journal of Statistical Mechanics Theory and Experiment (Impact Factor: 2.4). 05/2006; 2006(10). DOI: 10.1088/1742-5468/2006/10/L10003
Source: arXiv


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    ABSTRACT: We theoretically study divergent fluctuations of dynamical events at non-ergodic transitions. We first focus on the finding that a non-ergodic transition can be described as a saddle connection bifurcation of an order parameter for a time correlation function. Then, following the basic idea of Ginzburg-Landau theory for critical phenomena, we construct a phenomenological framework with which we can determine the critical statistical properties at saddle connection bifurcation points. Employing this framework, we analyze a model by considering the fluctuations of an instanton in space-time configurations of the order parameter. We then obtain the exponents characterizing the divergences of the length scale, the time scale and the amplitude of the fluctuations of the order parameter at the saddle connection bifurcation. The results are to be compared with those of previous studies of the four-point dynamic susceptibility at non-ergodic transitions in glassy systems.
    EPL (Europhysics Letters) 02/2007; 77(5):50008. DOI:10.1209/0295-5075/77/50008 · 2.10 Impact Factor
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    ABSTRACT: We develop a field theoretical treatment of a model of interacting Brownian particles. We pay particular attention to the requirement of time reversal invariance and the fluctuation-dissipation relationship (FDR). The method used is a modified version of the auxiliary field method due originally to Andreanov, Biroli and Lefevre (2006 J. Stat. Mech. P07008). We recover the correct diffusion law when the interaction is dropped as well as the standard mode coupling equation in the one-loop order calculation for interacting Brownian particle systems. It is noteworthy that despite our starting dynamical model containing the bare inter-particle interaction potential, it cancels out and the direct correlation function emerges in the end result due to our use of the correct static input.
    Journal of Statistical Mechanics Theory and Experiment 02/2008; 2008(02):P02004. DOI:10.1088/1742-5468/2008/02/P02004 · 2.40 Impact Factor
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    ABSTRACT: We demonstrate that scale-free patterns are observed in a spatially extended stochastic system whose deterministic part undergoes a saddle-node bifurcation. Remarkably, the scale-free patterns appear only at a particular time in relaxation processes from a spatially homogeneous initial condition. We characterize the scale-free nature in terms of the spatial configuration of the exiting time from a marginal saddle where the pair annihilation of a saddle and a node occurs at the bifurcation point. Critical exponents associated with the scale-free patterns are determined by numerical experiments.
    Physical Review E 12/2008; 78(5 Pt 2):055202. DOI:10.1103/PhysRevE.78.055202 · 2.29 Impact Factor
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