Scaled Langevin equation to describe the 1/fαspectrum

Based on an ideal system under external random forces, the dynamical process of random activation is studied. The time evolution of the system is described by the Langevin equation, and a scaling rule is introduced to generalize the system. The generalized system predicts the fractional power spectrum 1/${\mathit{f}}^{\mathrm{\alpha }}$ from a white spectrum to a Lorentzian. The exponent α is a function of the fractal dimension of the scaling rule. It is found that the fractal dimensions of 2, 1, and about 0.47 specify the particular mode of the generalized system, where the total power of the fractional power spectrum is minimum. The values indicate a Lorentz spectrum, a 1/f spectrum, and a power spectrum of 1/${\mathit{f}}^{1.53}$ type, respectively. The system of the minimum total power in this study is equivalent to one in the minimum potential energy, where the system is in the steady state. Therefore the random-activated system in the steady state gives a 1/f spectrum, and a 1/${\mathit{f}}^{\mathrm{\alpha }}$ spectrum is considered to represent the fluctuation of the complex system from the steady state.