Article

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

Journal of Statistical Mechanics Theory and Experiment (Impact Factor: 1.87). 05/2006; DOI: 10.1088/1742-5468/2006/10/L10003

Source: arXiv

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**ABSTRACT:**A Langevin equation whose deterministic part undergoes a saddle-node bifurcation is investigated theoretically. It is found that statistical properties of relaxation trajectories in this system exhibit divergent behaviors near a saddle-node bifurcation point in the weak-noise limit, while the final value of the deterministic solution changes discontinuously at the point. A systematic formulation for analyzing a path probability measure is constructed on the basis of a singular perturbation method. In this formulation, the critical nature turns out to originate from the neutrality of exiting time from a saddle point. The theoretical calculation explains results of numerical simulations.Physical Review E 07/2010; 82(1 Pt 1):011127. · 2.31 Impact Factor - [Show abstract] [Hide abstract]

**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. · 2.31 Impact Factor - [Show abstract] [Hide abstract]

**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. · 1.87 Impact Factor

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