Eigenvalue method to compute the largest relaxation time of disordered systems TeX Export

@article{citeulike:6014647, abstract = {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\$.}, archivePrefix = {arXiv}, author = {Monthus, Cecile and Garel, Thomas}, citeulike-article-id = {6014647}, citeulike-linkout-0 = {http://arxiv.org/abs/0910.4833}, citeulike-linkout-1 = {http://arxiv.org/pdf/0910.4833}, day = {26}, eprint = {0910.4833}, keywords = {detailed\_balance, disorder, eigenvalue, relaxation\_time, sherrington\_kirkpatrick}, month = {Oct}, posted-at = {2009-10-27 08:20:44}, priority = {2}, title = {Eigenvalue method to compute the largest relaxation time of disordered systems}, url = {http://arxiv.org/abs/0910.4833}, year = {2009} }

TY - JOUR ID - citeulike:6014647 L3 - citeulike-article-id:6014647 N2 - 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 $\eta \simeq 1.36$. TI - Eigenvalue method to compute the largest relaxation time of disordered systems KW - detailed_balance KW - disorder KW - eigenvalue KW - relaxation_time KW - sherrington_kirkpatrick AU - Monthus, Cecile AU - Garel, Thomas PY - 2009/Oct/26/ UR - http://arxiv.org/abs/0910.4833 ER -