1/<i>f</i>noise in percolation and percolationlike systems

Abstract

The behavior of 1/f noise effective intensity in two-phase percolation systems and percolationlike systems with an exponentially wide distribution of bond resistances is reviewed. Monte Carlo simulations on random resistor networks are performed. For a two-phase system the numerical values of noise critical exponents \ensuremath{\kappa}=1.54\ifmmode\pm\else\textpm\fi{}0.025, \ensuremath{\kappa}\ensuremath{'}=0.61\ifmmode\pm\else\textpm\fi{}0.02, w=6.31\ifmmode\pm\else\textpm\fi{}0.25, and w\ensuremath{'}=6.9\ifmmode\pm\else\textpm\fi{}0.25 are found in agreement with theoretical analysis performed with the help of a hierarchical model of a two-phase percolation system. For a system with an exponentially wide spectrum of bond resistances, i.e., a system in which bonds take on resistances r=${\mathit{r}}_{0}$ exp(-\ensuremath{\lambda}x), where \ensuremath{\lambda}\ensuremath{\gg}1 and x is a random variable, it is assumed that in the individual resistors the noise generating mechanism obeys the form {\ensuremath{\delta}${\mathit{r}}^{2}$}\ensuremath{\sim}${\mathit{r}}^{2+\mathrm{\ensuremath{\theta}}}$. In this case the effective noise intensity ${\mathit{C}}_{\mathit{e}}$\ensuremath{\equiv}S\ensuremath{\Omega}, where S is the relative power spectral density of system resistance fluctuations and \ensuremath{\Omega} is the system volume, is given by ${\mathit{C}}_{\mathit{e}}$\ensuremath{\sim}${\ensuremath{\lambda}}^{\mathit{m}}$ exp(-\ensuremath{\lambda}\ensuremath{\theta}${\mathit{x}}_{\mathit{c}}$), where 1-${\mathit{x}}_{\mathit{c}}$ is the percolation threshold. The exponent m is ``double universal,'' i.e., it is independent of lattice geometry and of the microscopic noise generating mechanism. Numerical simulations performed for \ensuremath{\theta}=1 and 0 give approximately m\ensuremath{\simeq}2.3 and confirm this ``double universality'' of the exponent m. The connections between 1/f noise effective intensity and effective susceptibility in a two-phase weakly nonlinear percolation system are also established. \textcopyright{} 1996 The American Physical Society.