Critical Behaviors and Finite-Size Scaling of Principal Fluctuation Modes in Complex Systems |
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Authors: | Xiao-Teng Li Xiao-Song Chen |
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Institution: | Institute of Theoretical Physics, Key Laboratory of Theoretical Physics, Chinese Academy of Sciences, P. O. Box 2735, Beijing 100190, China;
School of Physical Sciences, University of Chinese Academy of Sciences, No. 19A Yuquan Road, Beijing 100049, China |
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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)=Lγ/ν fn(tL1/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)=Lξ(tL1/ν). It is shown that the second moment correlation length in the two-dimensional square lattice is anisotropic. |
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Keywords: | critical phenomena finite-size scaling principal fluctuation modes |
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