Universal level dynamics of complex systems.
We study the evolution of the distribution of eigenvalues of a N x N matrix subject to a random perturbation drawn from (i) a generalized Gaussian ensemble and (ii) a non-Gaussian ensemble with a measure variable under the change of basis. It turns out that, in case (i), a redefiniton of the parameter governing the evolution leads to a Fokker-Planck equation similar to the one obtained when the perturbation is taken from a standard Gaussian ensemble (with invaraiant measure). This equivalence can therefore help us to obtain the correlations for various physically significant cases modeled by generalized Gaussian ensembles by using the already known correlations for standard Gaussian ensembles. For large N values, our results for both cases (i) and (ii) are similar to those obtained for the Wigner-Dyson gas as well as for the perturbation taken from a standard Gaussian ensemble. This seems to suggest the independence of evolution, in the thermodynamic limit, from the nature of perturbation involved as well as the initial conditions and therefore the universality of dynamics of the eigenvalues of complex systems.
Shukla
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Department of Physics, Indian Institute of Science, Bangalore 560012, India. Shukla@phy.iitkgp.ernet.in