Eigenvalue method to compute the largest relaxation time of disordered systems

Details Eigenvalue method to compute the largest relaxation time of disordered systems Eigenvalue method to compute the largest relaxation time of disordered systems

2009-10-26

arxiv.org/abs/0910.4833v2

J. Stat. Mech. (2009) P12017

Physics

Condensed Matter

Disordered Systems and Neural Networks

10 pages, 4 figures ; in v2, new rare-event interpretation of the tail exponent $\eta$ in relation with the sample-to-sample f

Scientific paper

10.1088/1742-5468/2009/12/P12017

We consider the dynamics of finite-size disordered systems as defined by a master equation satisfying detailed balance. The master equation can be mapped onto a Schr\"odinger equation in configuration space, where the quantum Hamiltonian $H$ has the generic form of an Anderson localization tight-binding model. The largest relaxation time $t_{eq}$ governing the convergence towards Boltzmann equilibrium is determined by the lowest non-vanishing eigenvalue $E_1=1/t_{eq}$ of $H$ (the lowest eigenvalue being $E_0=0$). So the relaxation time $t_{eq}$ can be computed {\it without simulating the dynamics} by any eigenvalue method able to compute the first excited energy $E_1$. Here we use the 'conjugate gradient' method to determine $E_1$ in each disordered sample and present numerical results on the statistics of the relaxation time $t_{eq}$ over the disordered samples of a given size for two models : (i) for the random walk in a self-affine potential of Hurst exponent $H$ on a two-dimensional square of size $L \times L$, we find the activated scaling $\ln t_{eq}(L) \sim L^{\psi}$ with $\psi=H$ as expected; (ii) for the dynamics of the Sherrington-Kirkpatrick spin-glass model of $N$ spins, we find the growth $\ln t_{eq}(N) \sim N^{\psi}$ with $\psi=1/3$ in agreement with most previous Monte-Carlo measures. In addition, we find that the rescaled distribution of $(\ln t_{eq})$ decays as $e^{- u^{\eta}}$ for large $u$ with a tail exponent of order $\eta \simeq 1.36$. We give a rare-event interpretation of this value, that points towards a sample-to-sample fluctuation exponent of order $\psi_{width} \simeq 0.26$ for the barrier.

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