Negative Specific Heat of a Magnetically Self-Confined Plasma Torus

It is shown that the thermodynamic maximum entropy principle predicts negative specific heat for a stationary magnetically self-confined current-carrying plasma torus. Implications for the magnetic self-confinement of fusion plasma are considered.Comment: 10p., LaTeX, 2 eps figure file

The Dirac point electron in zero-gravity Kerr--Newman spacetime

Dirac's wave equation for a point electron in the topologically nontrivial maximal analytically extended electromagnetic Kerr--Newman spacetime is studied in a zero-gravity limit; here, "zero-gravity" means $G\to 0$, where $G$ is Newton's constant of universal gravitation. The following results are obtained: the formal Dirac Hamiltonian on the static spacelike slices is essentially self-adjoint; the spectrum of the self-adjoint extension is symmetric about zero, featuring a continuum with a gap about zero that, under two smallness conditions, contains a point spectrum. Some of our results extend to a generalization of the zero-$G$ Kerr--Newman spacetime with different electric-monopole-to-magnetic-dipole-moment ratio.Comment: 49 pages, 17 figures; referee's comments implemented; the endnotes in the published version appear as footnotes in this preprin

The Vlasov continuum limit for the classical microcanonical ensemble

For classical Hamiltonian N-body systems with mildly regular pair interaction potential it is shown that when N tends to infinity in a fixed bounded domain, with energy E scaling quadratically in N proportional to e, then Boltzmann's ergodic ensemble entropy S(N,E) has the asymptotic expansion S(N,E) = - N log N + s(e) N + o(N); here, the N log N term is combinatorial in origin and independent of the rescaled Hamiltonian while s(e) is the system-specific Boltzmann entropy per particle, i.e. -s(e) is the minimum of Boltzmann's H-function for a perfect gas of "energy" e subjected to a combination of externally and self-generated fields. It is also shown that any limit point of the n-point marginal ensemble measures is a linear convex superposition of n-fold products of the H-function-minimizing one-point functions. The proofs are direct, in the sense that (a) the map E to S(E) is studied rather than its inverse S to E(S); (b) no regularization of the microcanonical measure Dirac(E-H) is invoked, and (c) no detour via the canonical ensemble. The proofs hold irrespective of whether microcanonical and canonical ensembles are equivalent or not.Comment: Final version; a few typos corrected; minor changes in the presentatio

The linear Fokker-Planck equation for the Ornstein-Uhlenbeck process as an (almost) nonlinear kinetic equation for an isolated N-particle system

It is long known that the Fokker-Planck equation with prescribed constant coefficients of diffusion and linear friction describes the ensemble average of the stochastic evolutions in velocity space of a Brownian test particle immersed in a heat bath of fixed temperature. Apparently, it is not so well known that the same partial differential equation, but now with constant coefficients which are functionals of the solution itself rather than being prescribed, describes the kinetic evolution (in the infinite particle limit) of an isolated N-particle system with certain stochastic interactions. Here we discuss in detail this recently discovered interpretation.Comment: Minor revisions and corrections (including the title

Misleading signposts along the de Broglie-Bohm road to quantum mechanics

Eighty years after de Broglie's, and a little more than half a century after Bohm's seminal papers, the de Broglie--Bohm theory (a.k.a. Bohmian mechanics), which is presumably the simplest theory which explains the orthodox quantum mechanics formalism, has reached an exemplary state of conceptual clarity and mathematical integrity. No other theory of quantum mechanics comes even close. Yet anyone curious enough to walk this road to quantum mechanics is soon being confused by many misleading signposts that have been put up, and not just by its detractors, but unfortunately enough also by some of its proponents. This paper outlines a road map to help navigate ones way.Comment: Dedicated to Jeffrey Bub on occasion of his 65th birthday. Accepted for publication in Foundations of Physics. A "slip of pen" in the bibliography has been corrected -- thanks go to Oliver Passon for catching it

Nonperturbative calculation of Born-Infeld effects on the Schroedinger spectrum of the hydrogen atom

We present the first nonperturbative numerical calculations of the nonrelativistic hydrogen spectrum as predicted by first-quantized electrodynamics with nonlinear Maxwell-Born-Infeld field equations. We also show rigorous upper and lower bounds on the ground state. When judged against empirical data our results significantly restrict the range of viable values of the new electromagnetic constant which is introduced by the Born-Infeld theory. We assess Born's own proposal for the value of his constant.Comment: 4p., 2 figs, 1 table; submitted for publicatio

The Vlasov limit and its fluctuations for a system of particles which interact by means of a wave field

In two recent publications [Commun. PDE, vol.22, p.307--335 (1997), Commun. Math. Phys., vol.203, p.1--19 (1999)], A. Komech, M. Kunze and H. Spohn studied the joint dynamics of a classical point particle and a wave type generalization of the Newtonian gravity potential, coupled in a regularized way. In the present paper the many-body dynamics of this model is studied. The Vlasov continuum limit is obtained in form equivalent to a weak law of large numbers. We also establish a central limit theorem for the fluctuations around this limit.Comment: 68 pages. Smaller corrections: two inequalities in sections 3 and two inequalities in section 4, and definition of a Banach space in appendix A1. Presentation of LLN and CLT in section 4.3 improved. Notation improve

On the Quasi-Linear Elliptic PDE −∇⋅(∇u/1−∣∇u∣2)=4π∑kakδsk-\nabla\cdot(\nabla{u}/\sqrt{1-|\nabla{u}|^2}) = 4\pi\sum_k a_k \delta_{s_k} in Physics and Geometry

It is shown that for each finite number of Dirac measures supported at points $s_n$ in three-dimensional Euclidean space, with given amplitudes $a_n$, there exists a unique real-valued Lipschitz function $u$, vanishing at infinity, which distributionally solves the quasi-linear elliptic partial differential equation of divergence form $-\nabla\cdot(\nabla{u}/\sqrt{1-|\nabla{u}|^2})=4\pi\sum_{n=1}^N a_n \delta_{s_n}$. Moreover, $u$ is real analytic away from the $s_n$. The result can be interpreted in at least two ways: (a) for any number of point charges of arbitrary magnitude and sign at prescribed locations $s_n$ in three-dimensional Euclidean space there exists a unique electrostatic field which satisfies the Maxwell-Born-Infeld field equations smoothly away from the point charges and vanishes as $|s|\to\infty$; (b) for any number of integral mean curvatures assigned to locations $s_n$ there exists a unique asymptotically flat, almost everywhere space-like maximal slice with point defects of Minkowski spacetime, having lightcone singularities over the $s_n$ but being smooth otherwise, and whose height function vanishes as $|s|\to\infty$. No struts between the point singularities ever occur.Comment: This is the preprint of the version published in 2012 in Commun. Math. Phys. PLUS an errata which has been accepted 08/13/2018 for publication in Commun. Math. Phy

Mean Field Theory of Spherical Gravitating Systems

Important gaps remain in our understanding of the thermodynamics and statistical physics of self-gravitating systems. Using mean field theory, here we investigate the equilibrium properties of several spherically symmetric model systems confined in a finite domain consisting of either point masses, or rotating mass shells of different dimension. We establish a direct connection between the spherically symmetric equilibrium states of a self-gravitating point mass system and a shell model of dimension 3. We construct the equilibrium density functions by maximizing the entropy subject to the usual constraints of normalization and energy, but we also take into account the constraint on the sum of the squares of the individual angular momenta, which is also an integral of motion for these symmetric systems. Two new statistical ensembles are introduced which incorporate the additional constraint. They are used to investigate the possible occurrence of a phase transition as the defining parameters for each ensemble are altered