Abstract: This is a pedagogical introduction to Brownian motion on the occasion of the 100th anniversary of Einstein's 1905 paper on the subject. After briefly reviewing Einstein's work in a contemporary context, we pursue several lines of further developments and applications to soft condensed matter and biology. Over the last century Brownian motion has been promoted from an odd curiosity of marginal scientific interest to a guiding theme pervading all of the modern life sciences.

Abstract: Einstein’s 1905 paper on Brownian motion was an essential contribution to the foundation of modern atomism [20]. Atomism as understood in science today presupposes, like its predecessor rooted in the theories of nature from Greek antiquity and from early modern times, that matter is constituted by small entities. But it no longer assumes that the properties and the behavior of these entities can simply be inferred from the familiar physical laws governing our macroscopic environment, nor that a description of matter in terms of its atomistic constituents can be exhaustive. Einstein succeeded in interpreting the irregular movements of small particles suspended in a liquid as visible evidence for themolecularmotions constituting the heat of a ponderable body according to the kinetic theory of heat. But he did so by radically changing the understanding of these irregular motions which he no longer conceived as being characterized by a velocity in the classical sense but as a stochastic process that can only be described with the help of statistical methods. It is therefore not surprising that Einstein’s work on Brownian motion also became one of the pillars of modern statistical thermodynamics and, more generally, of the physics of stochastic processes. In the sequel to his groundbreaking work, Einstein published several other related articles, extending the subject to Brownian motion in condensers and the fluctuations of heat radiation. His work aroused widespread interest among physicists and chemists, as indicated by Einstein’s correspondence with other scientists interested in the subject, in particular Conrad Rontgen, Richard Lorenz, Marian von Smoluchowski, and The Svedberg. In 1906 the Polish physicist von Smoluchowski submitted a paper on the kinetic theory of Brownian motion to the Annalen that was stimulated by Einstein’s papers but represented results which he had derived independently. While Smoluchowski’s argument was different from Einstein’s, his results were – apart from a numerical factor – essentially equivalent. Einstein’s interpretation of Brownian motion soon also received striking experimental confirmation by Jean Perrin and others. This success furthered the general acceptance of atomism and helped to convert the then still numerous skeptics. Indeed, while in the nineteenth century atomismwas widely employed as a working hypothesis in numerous fields of physics and chemistry, it was accepted as a physical reality only after the impressive accumulation

Abstract: In previous work it has been shown that the electromagnetic quantum vacuum, or electromagnetic zero-point field, makes a contribution to the inertial reaction force on an accelerated object. We show that the result for inertial mass can be extended to passive gravitational mass. As a consequence the weak equivalence principle, which equates inertial to passive gravitational mass, appears to be explainable. This in turn leads to a straightforward derivation of the classical Newtonian gravitational force. We call the inertia and gravitation connection with the vacuum fields the quantum vacuum inertia hypothesis. To date only the electromagnetic field has been considered. It remains to extend the hypothesis to the effects of the vacuum fields of the other interactions. We propose an idealized experiment involving a cavity resonator which, in principle, would test the hypothesis for the simple case in which only electromagnetic interactions are involved. This test also suggests a basis for the free parameter η(ν) which we have previously defined to parametrize the interaction between charge and the electromagnetic zero-point field contributing to the inertial mass of a particle or object.

Abstract: The aim of the present article is to study in detail the so-called spin equation (SE) and present both the methods of generating new solution and a new set of exact solutions. We recall that the SE with a real external field can be treated as a reduction of the Pauli equation to the (0 + 1)-dimensional case. Two-level systems can be described by an SE with a particular form of the external field. In this article, we also consider associated equations that are equivalent or (in one way or another) related to the SE. We describe the general solution of the SE and solve the inverse problem for this equation. We construct the evolution operator for the SE and consider methods of generating new sets of exact solutions. Finally, we find a new set of exact solutions of the SE.

Abstract: We consider spacetime to be a connected real 4-manifold equipped with a Lorentzian metric and an affine connection. The 10 independent components of the (symmetric) metric tensor and the 64 connection coefficients are the unknowns of our theory. We introduce an action which is (purely) quadratic in curvature and study the resulting system of Euler-Lagrange equations. In the first part of the paper we look for Riemannian solutions, i.e. solutions whose connection is Levi-Civita. We find two classes of Riemannian solutions: 1) Einstein spaces, and 2) spacetimes with pp-wave metric of parallel Ricci curvature. We prove that for a generic quadratic action these are the only Riemannian solutions. In the second part of the paper we look for non-Riemannian solutions. We define the notion of a "Weyl pseudoinstanton" (metric compatible spacetime whose curvature is purely of Weyl type) and prove that a Weyl pseudoinstanton is a solution of our field equations. Using the pseudoinstanton approach we construct explicitly a non-Riemannian solution which is a wave of torsion in a spacetime with Minkowski metric. We discuss the possibility of using this non-Riemannian solution as a mathematical model for the neutrino. (c) 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Abstract: The axioms of topological electromagnetism that were given by Hehl, Obukhov, and Rubilar are refined by the use of geometrical and topological notions that are found on orientable manifolds. The central problem of defining the spacetime electromagnetic constitutive law in terms of the geometrical and topological structure of the spacetime manifold is elaborated upon in the linear and nonlinear cases. The manner by which the spacetime metric might follow from the electromagnetic constitutive law is examined in the linear case. The possibility that the intersection form of the spacetime manifold might play a role in defining a topological basis for a nonlinear electromagnetic constitutive law is explored. The manner by which electromagnetic wave motion relates to the geometric structure is also discussed.

Abstract: We calculate the quantum phase transition for a homogeneous Bose gas in the plane of s-wave scattering length as and temperature T. This is done by improving a one-loop result near the interaction-free Bose-Einstein critical temperature Tc(0) with the help of recent high-loop results on the shift of the critical temperature due to a weak atomic repulsion based on variational perturbation theory. The quantum phase diagram shows a nose above Tc(0), so that we predict the existence of a reentrant transition above Tc(0), where an increasing repulsion leads to the formation of a condensate.

Abstract: We focus our attention, once again, on the Klein–Gordon and Dirac equations with a plane-wave field. We recall that for the first time a set of solutions of these equations was found by Volkov. The Volkov solutions are widely used in calculations of quantum effects with electrons and other elementary particles in laser beams. We demonstrate that one can construct sets of solutions which differ from the Volkov solutions and which may be useful in physical applications. For this purpose, we show that the transversal charge motion in a plane wave can be mapped by a special transformation to transversal free particle motion. This allows us to find new sets of solutions where the transversal motion is characterized by quantum numbers different from Volkov’s (in the Volkov solutions this motion is characterized by the transversal momentum). In particular, we construct solutions with semiclassical transversal charge motion (transversal squeezed coherent states). In addition, we demonstrate how the plane-wave field can be eliminated from the transversal charge motion in a more complicated case of the so-called combined electromagnetic field (a combination of a plane-wave field and constant colinear electric and magnetic fields). Thus, we find new sets of solutions of the Klein–Gordon and Dirac equations with the combined electromagnetic field.

Abstract: It is proved that classical BRS-invariance of the Lagrangian implies perturbative gauge invariance for tree diagrams to all orders. The proof applies in particular to the Einstein Hilbert Lagrangian of gravity.

Abstract: The observation of superluminal transmission of photons created discussions of their reality and significance. To analyze the possibility of such phenomena in general, the most general method is employed based on the covariant theory of dispersion relations. It is shown that superluminal phenomena can only be realized as tunneling of excitations within the near field zone. On this basis, the main experimental data can be described as sequential processes of single photon scattering. Their temporal distribution is estimated by taking into account the scattering time. Thus, the difference is explained between the advance of the superluminal part of the light pulse (i.e., the transition time of single photons) and the signal velocity of the macroscopic pulse of light.

Abstract: Using a point-charge calculation of the electrostatic crystal field, we determine the non-degenerate orbital ground state of the ferromagnetic Mott insulator YTiO3, which is found to agree perfectly with experiment. Based on the orbital order, we obtain by perturbation theory an effective spin Hamiltonian that describes the magnetic superexchange between nearest-neighbor Ti ions. The superexchange Hamiltonian includes, in addition to the isotropic Heisenberg coupling, antisymmetric (Dzyaloshinskii-Moriya) and symmetric anisotropy terms, caused by the spin-orbit interaction on the Ti ions. We find ferromagnetic Heisenberg couplings for Ti–Ti bonds in the crystallographic ab planes, but antiferromagnetic ones for Ti–Ti bonds between planes, in contradiction with experiment (which gives ferromagnetic couplings for both). Difficulties in calculating realistic values for the isotropic couplings of YTiO3 have been already reported in the literature. We discuss possible origins for these discrepancies. However, the much smaller values we obtain for the symmetric and antisymmetric anisotropies may be expected to be reliable. We therefore combine the experimentally-deduced isotropic coupling with the calculated anisotropic ones to determine the magnetic order of the Ti ions, which is found to be in satisfactory agreement with experiment. Based on this magnetic order, we derive the spin-wave spectrum. We find an acoustic branch with a very small zone-center gap and three optical spin-wave modes with sizeable zone-center gaps. The acoustic branch reproduces the one reported in experiment, and the optical ones are in a satisfactory agreement with experiment, upon a proper folding of the magnetic Brillouin zone.

Abstract: Magnetic impurities in small metallic clusters are investigated in the framework of the Anderson model by using exact diagonalization and geometry optimization methods.The singlet-triplet spin gap ΔE shows a remarkable dependence as a function of band-filling, cluster structure, and impurity position that can be interpreted in terms of the environment-specific conduction-electron spectrum. The low-energy spin excitations involve similar energies as isomerizations. Interesting correlations between cluster structure and magnetic behavior are revealed. Finite-temperature properties such as specific heat, effective impurity moment, and magnetic susceptibility are calculated exactly in the canonical ensemble. A finite-size equivalent of the Kondo effect is identified and its structural dependence is discussed.

Abstract: We investigate (non-relativistic) atomic systems interacting with quantum electromagnetic field (QEF). The resulting model describes spontaneous emission of light from a two-level atom surrounded by various initial states of the QEF. We assume that the quantum field interacts with the atom via the standard, minimal-coupling Hamiltonian, with the A2 term neglected. We also assume that there will appear at most single excitations (photons). By conducting the analysis on a general level we allow for an arbitrary initial state of the QEF (which can be for instance: the vacuum, the ground state in a cavity, or the squeezed state). We derive a Volterra-type equation which governs the time evolution of the amplitude of the excited state. The two-point function of the initial state of the QEF, integrated with a combination of atomic wavefunctions, forms the kernel of this equation.