Asymptotics of some generalized Mathieu series

We establish asymptotic estimates of Mathieu-type series defined by sequences with power-logarithmic or factorial behavior. By taking the Mellin transform, the problem is mapped to the singular behavior of certain Dirichlet series, which is then translated into asymptotics for the original series. In the case of power-logarithmic sequences, we obtain precise first order asymptotics. For factorial sequences, a natural boundary of the Mellin transform makes the problem more challenging, but a direct elementary estimate gives reasonably precise asymptotics. As a byproduct, we prove an expansion of the functional inverse of the gamma function at infinity.

Dynamic properties of ηi-mode weak turbulence

The turbulence of the toroidal ηi mode has a forward spectral cascade. This means that most of the energy initially placed in the low-wavenumber region (the linear instability region) will be spread out towards high-wavenumber modes. Therefore the linear instability may be reduced by this energy cascade. An elementary estimate of the critical amplitude for nonlinear mode coupling to balance the linear growth rate is given. As a consequence of three wave cascade process derivable from the fully toroidal fluid model equations, the formation of zonal flows ηi-mode turbulence is predicted.

METAL-SURFACE CORRELATION ENERGY FROM THE LIQUID DROP MODEL: A BACK-OF-THE-ENVELOPE ESTIMATE

The liquid drop model applied to the one-electron problem provides an elementary estimate of the correlation contribution to the surface and curvature energies of jellium, in terms of bulk electron density and bulk correlation energy. Within the random phase approximation (RPA), this estimate correctly predicts the size of the surface correlation energy, its strong dependence upon bulk density, and its weak dependence upon surface density profile. The local density approximation (LDA) to RPA predicts surface correlation energies that are far too small, as a consequence of the LDA self-interaction error. Possible implications beyond RPA are discussed. The power and limitations of the liquid drop expansion are illustrated by the example of one-electron jellium spheroids.