Ground State H-Atom in Born-Infeld Theory

Within the context of Born-Infeld (BI) nonlinear electrodynamics (NED) we revisit the non-relativistic, spinless H-atom. The pair potential computed from the Born-Infeld equations is approximated by the Morse type potential with remarkable fit over the critical region where the convergence of both the short and long distance expansions slows down dramatically. The Morse potential is employed to determine both the ground state energy of the electron and the BI parameter.Comment: 4 pages, 1 figure, final version to appear in Foundation of Physic

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

The Mean-Field Limit for a Regularized Vlasov-Maxwell Dynamics

The present work establishes the mean-field limit of a N-particle system towards a regularized variant of the relativistic Vlasov-Maxwell system, following the work of Braun-Hepp [Comm. in Math. Phys. 56 (1977), 101-113] and Dobrushin [Func. Anal. Appl. 13 (1979), 115-123] for the Vlasov-Poisson system. The main ingredients in the analysis of this system are (a) a kinetic formulation of the Maxwell equations in terms of a distribution of electromagnetic potential in the momentum variable, (b) a regularization procedure for which an analogue of the total energy - i.e. the kinetic energy of the particles plus the energy of the electromagnetic field - is conserved and (c) an analogue of Dobrushin's stability estimate for the Monge-Kantorovich-Rubinstein distance between two solutions of the regularized Vlasov-Poisson dynamics adapted to retarded potentials.Comment: 34 page

Monotonicity of quantum ground state energies: Bosonic atoms and stars

The N-dependence of the non-relativistic bosonic ground state energy is studied for quantum N-body systems with either Coulomb or Newton interactions. The Coulomb systems are "bosonic atoms," with their nucleus fixed, and the Newton systems are "bosonic stars". In either case there exists some third order polynomial in N such that the ratio of the ground state energy to the respective polynomial grows monotonically in N. Some applications of these new monotonicity results are discussed

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

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

Order and Chaos in some Trigonometric Series: Curious Adventures of a Statistical Mechanic

This paper tells the story how a MAPLE-assisted quest for an interesting undergraduate problem in trigonometric series led some "amateurs" to the discovery that the one-parameter family of deterministic trigonometric series \pzcS_p: t\mapsto \sum_{n\in\Nset}\sin(n^{-{p}}t), $p>1$, exhibits both order and apparent chaos, and how this has prompted some professionals to offer their expert insights. It is proved that \pzcS_p(t) = \alpha_p\rm{sign}(t)|t|^{1/{p}}+O(|t|^{1/{(p+1)}})\;\forall\;t\in\Rset, with explicitly computed constant $\alpha_p$. Experts' commentaries are reproduced stating the fluctuations of \pzcS_p(t) - \alpha_p{\rm{sign}}(t)|t|^{1/{p}} are presumably not Gaussian. Inspired by a central limit type theorem of Marc Kac, a well-motivated conjecture is formulated to the effect that the fluctuations of the $\lceil t^{1/(p+1)}\rceil$-th partial sum of \pzcS_p(t), when properly scaled, do converge in distribution to a standard Gaussian when $t\to\infty$, though --- provided that $p$ is chosen so that the frequencies \{n^{-p}\}_{n\in\Nset} are rationally linear independent; no conjecture has been forthcoming for rationally dependent \{n^{-p}\}_{n\in\Nset}. Moreover, following other experts' tip-offs, the interesting relationship of the asymptotics of \pzcS_p(t) to properties of the Riemann $\zeta$ function is exhibited using the Mellin transform.Comment: Based on the invited lecture with the same title delivered by the author on Dec.19, 2011 at the 106th Statistical Mechanics Meeting at Rutgers University in honor of Michael Fisher, Jerry Percus, and Ben Widom. (19 figures, colors online). Comments of three referees included. Conjecture 1 revised. Accepted for publication in J. Stat. Phy

Ground state at high density

Weak limits as the density tends to infinity of classical ground states of integrable pair potentials are shown to minimize the mean-field energy functional. By studying the latter we derive global properties of high-density ground state configurations in bounded domains and in infinite space. Our main result is a theorem stating that for interactions having a strictly positive Fourier transform the distribution of particles tends to be uniform as the density increases, while high-density ground states show some pattern if the Fourier transform is partially negative. The latter confirms the conclusion of earlier studies by Vlasov (1945), Kirzhnits and Nepomnyashchii (1971), and Likos et al. (2007). Other results include the proof that there is no Bravais lattice among high-density ground states of interactions whose Fourier transform has a negative part and the potential diverges or has a cusp at zero. We also show that in the ground state configurations of the penetrable sphere model particles are superposed on the sites of a close-packed lattice.Comment: Note adde

A dynamical classification of the range of pair interactions

We formalize a classification of pair interactions based on the convergence properties of the {\it forces} acting on particles as a function of system size. We do so by considering the behavior of the probability distribution function (PDF) P(F) of the force field F in a particle distribution in the limit that the size of the system is taken to infinity at constant particle density, i.e., in the "usual" thermodynamic limit. For a pair interaction potential V(r) with V(r) \rightarrow \infty) \sim 1/r^a defining a {\it bounded} pair force, we show that P(F) converges continuously to a well-defined and rapidly decreasing PDF if and only if the {\it pair force} is absolutely integrable, i.e., for a > d-1, where d is the spatial dimension. We refer to this case as {\it dynamically short-range}, because the dominant contribution to the force on a typical particle in this limit arises from particles in a finite neighborhood around it. For the {\it dynamically long-range} case, i.e., a \leq d-1, on the other hand, the dominant contribution to the force comes from the mean field due to the bulk, which becomes undefined in this limit. We discuss also how, for a \leq d-1 (and notably, for the case of gravity, a=d-2) P(F) may, in some cases, be defined in a weaker sense. This involves a regularization of the force summation which is generalization of the procedure employed to define gravitational forces in an infinite static homogeneous universe. We explain that the relevant classification in this context is, however, that which divides pair forces with a > d-2 (or a < d-2), for which the PDF of the {\it difference in forces} is defined (or not defined) in the infinite system limit, without any regularization. In the former case dynamics can, as for the (marginal) case of gravity, be defined consistently in an infinite uniform system.Comment: 12 pages, 1 figure; significantly shortened and focussed, additional references, version to appear in J. Stat. Phy