Statistics of the eigenvalues of Tsallis matrices

Quasistationarity in a model of long-range interacting particles moving on a sphere

“…The degree of clustering is conveniently measured by the "magnetization" vector m = (m x , m y , m z ) = (1/N ) N i=1 S i . In the BG equilibrium state, the system exhibits a continuous phase transition at the critical energy density ǫ c = 5/6, between a low-energy clustered ("magnetized") phase in which the particles are close together on the sphere, and a high-energy homogeneous ("non-magnetized") phase in which the particles are uniformly distributed on the sphere [37,38]. As a function of the energy, the magnitude of m, given by m = m 2…”

“…For example, it has been demonstrated that while anisotropic versions of the HMF model do exhibit QSSs [25], introducing stochastic processes into the dynamics tends to destroy QSSs leading to non-divergent relaxation times [31][32][33][34][35][36]. A very interesting generalization of the model to that of particles moving on the surface of a sphere rather than on a circle has recently been introduced [37,38]. Here, the model is defined on a larger phase space with each particle characterized by two positional degrees of freedom rather than one as is the case for the HMF model.…”

“…In equilibrium, the system exhibits a continuous phase transition from a lowenergy magnetized phase in which the particles are clustered on the spherical surface, across critical threshold energy ǫ c , to a high-energy homogeneous phase. Numerical studies of the model have shown that it exhibits a number of quasistationary states, with relaxation times which diverge with the system size [37,38]. It would be interesting to study analytically the relaxation processes in this higher dimensional model and trace the origin of its slow dynamics.…”

Nonextensive and superstatistical generalizations of random-matrix theory

“…where ε = (E 1 + E 2 )/2, s = |E 1 − E 2 |. For this case, the distribution (20) has to be complemented by the auxiliary condition that the quantity inside the square bracket has to be positive. We here consider the case of q ≥ 1 where no limitations are imposed on the values of the variables ε and s, and refer the reader interested in the other case of q < 1 to…”

Superstatistical random-matrix-theory approach to transition intensities in mixed systems

“…Recently, the formalism has been applied to include systems with mixed regular-chaotic dynamics in the framework of RMT [12,13,14,15,16,17]. However, the constraints of normalization and existence of an expectation value for Tr H † H set up an upper limit for the entropic index q beyond which the involved integrals diverge.…”