Critical Behaviors and Finite-Size Scaling of Principal Fluctuation Modes in Complex Systems*Supported by the National Natural Science Foundation of China under Grant Nos. 11121403 and 11504384. (1st September 2016) - Li 李, Xiao-Teng 晓腾; Chen 陈, Xiao-Song 晓松
Abstract: Complex systems consisting of N agents can be investigated from the aspect of principal fluctuation modes of agents. From the correlations between agents, an N × N correlation matrix C can be obtained. The principal fluctuation modes are defined by the eigenvectors of C. Near the critical point of a complex system, we anticipate that the principal fluctuation modes have the critical behaviors similar to that of the susceptibity. With the Ising model on a two-dimensional square lattice as an example, the critical behaviors of principal fluctuation modes have been studied. The eigenvalues of the first 9 principal fluctuation modes have been invesitigated. Our Monte Carlo data demonstrate that these eigenvalues of the system with size L and the reduced temperature t follow a finite-size scaling form λn (L, t) = L γ/ν fn (tL 1/ν ), where γ is critical exponent of susceptibility and ν is the critical exponent of the correlation length. Using eigenvalues λ1, λ2 and λ6, we get the finite-size scaling form of the second moment correlation length ξ ( L,   t ) = L ξ ˜ ( t L 1 / ν ) . It is shown that the second moment correlation length in the two-dimensional square lattice is anisotropic.