NUMERICAL APPROXIMATION OF THE SMOLUCHOWSKI EQUATION USING RADIAL BASIS FUNCTIONS

The goal of this paper is to present a numerical method for the Smoluchowski equation,a drift-diffusion equation on the sphere,arising in the modelling of particle dynamics.The numerical method uses radial basis functions(RBF).This is a relatively new approach,which has recently mainly been used for geophysical applications.For a simplified model problem we compare the RBF approach with a spectral method,i.e.the standard approach used in related physical applications.This comparison as well as our other accuracy studies show that RBF methods are an attractive alternative for these kind of models.

Optimal reactions for synthesis of nuclides of element 108

The cross sections for the production of nuclides of element 108 via hot fusion evaporation reactions are studied using a two-parameter Smoluchowski equation. The optimal reactions for the synthesis of new nuclides of element 108 with mass numbers from 266 to 271 are suggested. The macroscopic-microscopic approach predicts a strong deformed shell closure at Z ≈ 108 and N = 162. The synthesis of more nuclides of element 108 is meaningful to the confirmation of the existence of this deformed shell closure.

Observing dynamic oscillatory behavior of triple points among black hole thermodynamic phase transitions

Understanding the dynamic process of black hole thermodynamic phase transitions at a triple point is a huge challenge. In this paper, we conduct the first investigation of dynamic phase behavior at a black hole triple point. By numerically solving the Smoluchowski equation near the triple point for a six-dimensional charged Gauss-Bonnet anti-de Sitter black hole, we report that initial small, intermediate, or large black holes can transit to the other two coexistent phases at the triple point, indicating that thermodynamic phase transitions can indeed occur dynamically. More significantly, we observe characteristic weak and strong oscillatory behavior in this dynamic process, which can be understood from an investigation of the rate of first passage from one phase to another. Our results further an understanding of the dynamic process of black hole thermodynamic phase transitions.

ON THE NUMERICAL SOLUTION OF ELLIPTIC AND PARABOLIC PDE IN THE REAL PROJECTIVE PLANE

Some models dealing with fibers and liquid crystals can be formulated probabilistically in terms of orientation distributions. Since the orientation of a thin object can be specified by a point in a real projective plane this approach leads to elliptic and parabolic problems in the real projective plane. In most previous works these kind of problems have been considered on the unit sphere which is a double cover of the real projective plane. However, numerically this is inefficient because the resulting systems of equations are unnecessarily big. We formulate the problem directly in the real projective plane using a certain parametrization with three coordinate domains. After reducing the computations to the coordinate domains we can then use finite elements almost in a standard way. In particular the standard error estimates with usual Sobolev spaces remain valid in this setting. We consider both elliptic and parabolic cases, and demonstrate the validity of our approach.