Symmetry Results for Finite-Temperature,¶Relativistic Thomas-Fermi Equations

Abstract

In the semi-classical limit the relativistic quantum mechanics of a stationary beam of counter-streaming (negatively charged) electrons and one species of positively charged ions is described by a nonlinear system of finite-temperature Thomas-Fermi equations. In the high temperature / low density limit these Thomas-Fermi equations reduce to the (semi-)conformal system of Bennett equations discussed earlier by Lebowitz and the author. With the help of a sharp isoperimetric inequality it is shown that any hypothetical particle density function which is not radially symmetric about and decreasing away from the beam's axis would violate the virial theorem. Hence, all beams have the symmetry of the circular cylinder. * In celebration of the 70 th birthday of Joel L. Lebowitz.

Degree counting theorems for singular Liouville systems

“…1 35,36,37,38,39,40,41]). Various Liouville systems are also used to describe models in theories of chemotaxis ( [11,22]), the physics of charged particle beams [6,17,24,25], and other gauge field models [19,26]. Even if the system is reduced to one equation, it has profound background in geometry: if the equation has no singular source, it interprets the Nirenberg problem of prescribing Gauss curvature; if the equation has singular sources, the solution represents a metric with conic singularity [27].…”

On Non-Topological Solutions for Planar Liouville Systems of Toda-Type

“…The main purpose of this paper is to analyse the solvability of (not necessarily integrable) planar "'singular"' Liouville systems, as they occur in various Chern-Simons vortex problems, see e.g. [33,34,55], or in other physical context decribed for example in [44,45,46,47], and which include the Toda system as a particular case.…”

“…Actually, to carry out such "perturbation" approach, the first important step is to provide rather accurate informations about the solvability of Liouville -systems involving Dirac measures, which occur as "'limiting"' problems in the perturbation argument. Interestingly, the relevance of such class of Liouville systems has emerged already in several other contexts, see [13,44,45,46,47,21,43] and references therein. Their study has concerned mainly the so called "cooperative" case, where all the entries of the coupling matrix are assumed positive, and we refer to [12,13,21,63,64,70,52,53,54,58,59], for rather complete results about this situation.…”

On Liouville systems at critical parameters, Part 1: One bubble

“…In classical gauge field theory, equation (1.1) is closely related to the Chern-Simons-Higgs equation for the abelian case, see [5,24,25,45]. Various Liouville systems are also used to describe models in the theory of self-gravitating systems [1], Chemotaxis [17,27], in the physics of charged particle beams [4,20,29,30], in the non-abelian Chern-Simons-Higgs theory [21,26,45] and other gauge field models [22,23,31]. For recent developments of these subjects or related Liouville systems in more general settings, we refer the readers to [2,3,12,13,14,15,18,19,32,33,34,35,36,37,40,41,42,43,44,46,47] and the references therein.…”