Complex-valued Second Difference as a Measure of Stabilization of Complex Dissipative Statistical Systems: Girko Ensemble

A complex quantum system with energy dissipation is considered. The quantum Hamiltonians H belong the complex Ginibre ensemble. The complexvalued eigenenergies Zi are random variables. The second differences ∆ Zi are also complex-valued random variables. The second differences have their real and imaginary parts and also radii (moduli) and main arguments (angles). ForN=3 dimensional Ginibre ensemble the distributions of above random variables are provided whereas for genericN dimensional Ginibre ensemble second difference’s, radius’s and angle’s distributions are analytically calculated. The law of homogenization of eigenergies is formulated. The analogy of Wigner and Dyson of Coulomb gas of electric charges is studied. The stabilisation of system of electric charges is dealt with.