Zero order nonrelativistic states

In the present work, the general mathematical scheme of construction of the equilibrium statistical mechanics on the basis of an arbitrary definition of statistical entropy for two types of thermodynamic potential, the first and the second thermodynamic potentials, was proposed. As an example, we investigated the Tsallis and Boltzmann-Gibbs statistical entropies in the canonical and microcanonical ensembles. On the example of a nonrelativistic ideal gas, it was proven that the statistical mechanics based on the Tsallis entropy satisfies the requirements of the equilibrium thermodynamics only in the thermodynamic limit when the entropic index z is an extensive variable of state of the system. In this case the thermodynamic quantities of the Tsallis statistics belong to one of the classes of homogeneous functions of the first or zero orders. 

In the case of a relativistic system, as a first (and useful) approximation, the zero-order spectrum can be taken as the nonrelativistic one, with Hq defined explicifly as fhe Coulomb Hamiltonian. Then, the perturbation V is also written explicitly as the relativistic Breit-Pauli operators, and it is this perturbation that turns the initially discrete state into a resonance. For example, this type of advanced calculation, with multichannel coupling included, has been shown to explain quantitatively the positions and lifetimes of the relativistic levels of mefastable states in negative ions [90]. However, if the more accurate four-component relativistic Dirac treatment for each electron is invoked for cases of high effective nuclear charge, then the stability against autoionization implies not only the exclusion of components representing decay to... 

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