Long-term behavior of Hertzian chains between fixed walls is really equilibrium

We examine the long-term behavior of nonintegrable, energy-conserved, 1D systems of macroscopic grains interacting via a contact-only generalized Hertz potential and held between stationary walls. Existing dynamical studies showed the absence of energy equipartitioning in such systems, hence their long-term dynamics was described as quasi-equilibrium. Here, we show that these systems do in fact reach thermal equilibrium at sufficiently long times, as indicated by the calculated heat capacity. This phase is described by equilibrium statistical mechanics, opening up the possibility that the machinery of nonequilibrium statistical mechanics may be used to understand the behavior of these systems away from equilibrium.

Hertz potential formalism for force-free electrodynamics and its application to Brennan–Gralla–Jacobson solutions

The Hertz potential is a powerful tool for the source-free electrodynamics. Especially for the algebraically special spacetime background, the Hertz potential formalism simplifies the Maxwell equations quite much. In astrophysics, strong electric–magnetic field is very common. Force-free electrodynamics is a good approximation for strong enough electric–magnetic field compared to the inertial energy of the involved plasma. For example, the force-free model has been extensively used to describe the magnetosphere of stars in the universe. In this paper, we extend the Hertz potential formalism to the force-free electrodynamics. The Hertz potential formalism simplifies the force-free dynamical equations as much as that in the source-free case. As an application, we use the Hertz potential formalism to the Schwarzschild background. And the Brennan–Gralla–Jacobson solutions are recovered straightforwardly.

SELF-COUPLING OF EXCITED ATOMS IN CURVED SPACE–TIME

It is well known that an excited atom, placed near a boundary, such as a mirror, undergoes an energy shift due to its interaction with the reflected field. In this paper, we use a generalized Hertz potential to prove that a radiating dipole embedded in a continuously inhomogeneous medium also experiences a position-dependent self-interaction energy shift and a corresponding self-force. Consequently, an excited atom inside a cylindrical cavity embedded in a quasi-homogeneous gravitational field, which acts as an effective "soft" boundary, is shown to experience an effective gravitational acceleration dependent on the atomic quantum state. We predict that excited trapped atom interferometers will thus provide an unexpected tool for ground-based experimentation on radiation backscattering in a Schwarzschild background.

The relation between Maxwell, Dirac, and the Seiberg-Witten equations

We discuss unsuspected relations between Maxwell, Dirac, and the Seiberg-Witten equations. First, we present the Maxwell-Dirac equivalence (MDE) of the first kind. Crucial to that proposed equivalence is the possibility of solving for ψ (a representative on a given spinorial frame of a Dirac-Hestenes spinor field) the equation F=ψγ21ψ˜, where F is a given electromagnetic field. Such task is presented and it permits to clarify some objections to the MDE which claim that no MDE may exist because F has six (real) degrees of freedom and ψ has eight (real) degrees of freedom. Also, we review the generalized Maxwell equation describing charges and monopoles. The enterprise is worth, even if there is no evidence until now for magnetic monopoles, because there are at least two faithful field equations that have the form of the generalized Maxwell equations. One is the generalized Hertz potential field equation (which we discuss in detail) associated with Maxwell theory and the other is a (nonlinear) equation (of the generalized Maxwell type) satisfied by the 2-form field part of a Dirac-Hestenes spinor field that solves the Dirac-Hestenes equation for a free electron. This is a new result which can also be called MDE of the second kind. Finally, we use the MDE of the first kind together with a reasonable hypothesis to give a derivation of the famous Seiberg-Witten equations on Minkowski spacetime. A physical interpretation for those equations is proposed.

New theoretical and practical aspects of electromagnetic soundings at low induction numbers

This paper presents a theoretical yet practical study of electromagnetic (EM) soundings at low induction numbers for vertical and horizontal magnetic dipoles. The physical model is a heterogeneous half‐space with arbitrary vertical conductivity variations. The study comprises a novel approach for solving forward problems, analytical formulas for inversion, and a practical algorithm for recovering conductivity variations from field measurements. The basis of the theoretical approach is a series representation of the EM field in terms of ascending powers of frequency. At low induction numbers only two terms are required. When substituted into Maxwell's equations, one term in the series can be obtained in terms of the other. Furthermore, if the electrical conductivity varies only with depth, the imaginary part of the field can be obtained from its real part through a differential equation. The real part, which corresponds to zero frequency, plays the role of a distributed source for the frequency‐dependent imaginary part. In the case of vertical magnetic dipoles, the approach applies directly to the real and imaginary components of the magnetic field, while for horizontal dipoles one must use the Hertz potential, but the procedure is exactly the same. In each case this leads to a statement of the forward problem as the solution of a real differential equation. The solutions are integral expressions valid for arbitrary conductivity profiles. Assuming that these expressions represent integral equations for conductivity, analytical inverse formulas are derived for both vertical and horizontal dipoles. These formulas ensure a unique recovery of the conductivity profile under ideal conditions. An algorithm based on linear programming offers a variety of practical advantages for the inversion of field data. Numerical experiments and applications to field data illustrate the performance of the algorithm.