Dynamical Analysis in Complex Networks and Quantum Motions

Abstract

In this paper, we have studied seismic network. Abe and Suzuki investigate novel seismic network from a singular seismic time series by considering cell resolution and temporal causality. But their approach to construct network does not express the meaning of aftershock. Because, an aftershock is a smaller earthquake to occur after a previous large one. Therefore, we suggest new method to construct seismic network using relationship between aftershock and main earthquake. With the new method, we have examined some topological properties of the earthquake such as the mean degree, the characteristic path length, the clustering coefficient, the global efficiency, the hierarchy, and probability distribution. And the our results compare with Abe and Suzuki's method We have simulated dynamical phase transitions in a Boolean network with initial random connections. The nature of the phase transition is found numerically and analytically in two connecting probability density function. By using the noise intensity, we show that a critical value exists for the noise intensity. In addition, we find that the critical exponent of our simulation is similar to the theoretical result 1/2. Lastly, we have studied the non-Markovian Caldeira–Leggett master equation for the Brownian motion of a free particle. The Fokker–Planck equation with the effective potential in the long time limit contains the Markovian Klein–Kramers equation with the diffusion energy. We mainly analyze the quantum Brownian motion with the harmonic oscillation in the one-dimensional quantum space. By using the Wigner function technique from the non-Markovian Caldeira-Leggett equation, we calculate the velocity distribution function with the diffusion energy and the correlational function. Since such three correlational functions are considered as the exponential, Gaussian, and complementary error functions. The quantum force can be analyzed from the velocity distribution function. Particularly, the quantum force is found to be proportional to the angular frequency in the quantum limit and the steady regime, while the classical force is proportional to the temperature in the classical limit.