Relativistic kinetic theory of magnetoplasmas

Recently, an increasing interest in astrophysical as well as laboratory plasmas has been manifested in reference to the existence of relativistic flows, related in turn to the production of intense electric fields in magnetized systems. Such phenomena require their description in the framework of a consistent relativistic kinetic theory, rather than on relativistic MHD equations, subject to specific closure conditions. The purpose of this work is to apply the relativistic single-particle guiding-center theory developed by Beklemishev and Tessarotto, including the nonlinear treatment of small-wavelength EM perturbations which may naturally arise in such systems. As a result, a closed set of relativistic gyrokinetic equations, consisting of the collisionless relativistic kinetic equation, expressed in hybrid gyrokinetic variables, and the averaged Maxwell's equations, is derived for an arbitrary four-dimensional coordinate system.Comment: 6 pages, 1 figure. Contributed to the Proceedings of the 24th International Symposium on Rarefied Gas Dynamics, July 10-16, 2004 Porto Giardino Monopoli (Bari), Ital

Turing jumps through provability

Fixing some computably enumerable theory $T$, the Friedman-Goldfarb-Harrington (FGH) theorem says that over elementary arithmetic, each $\Sigma_1$ formula is equivalent to some formula of the form $\Box_T \varphi$ provided that $T$ is consistent. In this paper we give various generalizations of the FGH theorem. In particular, for $n>1$ we relate $\Sigma_{n}$ formulas to provability statements $[n]_T^{\sf True}\varphi$ which are a formalization of "provable in $T$ together with all true $\Sigma_{n+1}$ sentences". As a corollary we conclude that each $[n]_T^{\sf True}$ is $\Sigma_{n+1}$-complete. This observation yields us to consider a recursively defined hierarchy of provability predicates $[n+1]^\Box_T$ which look a lot like $[n+1]_T^{\sf True}$ except that where $[n+1]_T^{\sf True}$ calls upon the oracle of all true $\Sigma_{n+2}$ sentences, the $[n+1]^\Box_T$ recursively calls upon the oracle of all true sentences of the form $\langle n \rangle_T^\Box\phi$. As such we obtain a `syntax-light' characterization of $\Sigma_{n+1}$ definability whence of Turing jumps which is readily extended beyond the finite. Moreover, we observe that the corresponding provability predicates $[n+1]_T^\Box$ are well behaved in that together they provide a sound interpretation of the polymodal provability logic ${\sf GLP}_\omega$