Classical light dispersion theory in a regular lattice

We study the dynamics of an infinite regular lattice of classical charged oscillators. Each individual oscillator is described as a point particle subject to a harmonic restoring potential, to the retarded electromagnetic field generated by all the other particles, and to the radiation reaction expressed according to the Lorentz--Dirac equation. Exact normal mode solutions, describing the propagation of plane electromagnetic waves through the lattice, are obtained for the complete linearized system of infinitely many oscillators. At variance with all the available results, our method is valid for any values of the frequency, or of the ratio between wavelength and lattice parameter. A remarkable feature is that the proper inclusion of radiation reaction in the dynamics of the individual oscillators does not give rise to any extinction coefficient for the global normal modes of the lattice. The dispersion relations resulting from our solution are numerically studied for the case of a simple cubic lattice. New predictions are obtained in this way about the behavior of the crystal at frequencies near the proper oscillation frequency of the dipoles.Comment: 15 pages, 1 figure; typos correcte

Zur allgemeinen Theorie der halbgeordneten Räume

Foreword by K. Kopotun11Correspondence to: K. Kopotun, Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, Canada ^^IR3T 2N2. Email: [email protected] paper “On the general theory of semi-ordered spaces” (“Zur allgemeinen Theorie der halbgeordneten Räume”) was written by L.V. Kantorovich and G.R. Lorentz22Until 1946, G.G. (Georg Gunter) Lorentz was using the name Geogrij Rudolfovich (G.R.) Lorentz. sometime in 1937–1939, and this is the first time it appears in print.The following is a short history of this manuscript.In his letter to I.P. Natanson written on October 11, 1937, G.G. Lorentz mentioned a talk on joint work with L.V. Kantorovich that he gave at a Session on Functional Analysis in Moscow earlier that year. The records of the Academy of Sciences of USSR indicate that a Session on Functional Analysis took place in Moscow during September 27–29, 1937, and that G.R. Lorentz gave a talk “Topological theory of semi-ordered spaces” there, and that L.V. Kantorovich was speaking on “Theory of linear operations in semi-ordered spaces”.The manuscript “On the general theory of semi-ordered spaces” was found in the archives of L.V. Kantorovich. According to Vsevolod Leonidovich Kantorovich, L.V. Kantorovich’s son, it was submitted to Trudy Tomskogo Gosudarstvennogo Universiteta imeni V. V. Kuibysheva (Proceedings of Tomsk State University). The typed version33See www.math.ohio-state.edu/~nevai/LORENTZ/KANTOROVICH_LORENTZ_typed.pdf/. of the manuscript has a handwritten note by N. Romanov44N.P. Romanov (1907–1972) was a Professor at Tomsk University from 1935 until 1944. After 1944 he worked in Uzbekistan. His main area of research was Number Theory and Theory of Functions of Complex Variables. For more information see “Nikolaĭ Pavlovich Romanov (on the eightieth anniversary of his birth)”, Izv. Akad. Nauk UzSSR Ser. Fiz.-Mat. Nauk 1987, no. 3, 92–93, MR0914654 (89b:01069). dated by August 31, 1939 stating that the manuscript is accepted for publication. The manuscript was never published (probably because of the World War II) and around 1945 was returned to L.V. Kantorovich.It has been decided to publish this manuscript in its original language (German), and translate the extended abstract accompanying this manuscript from Russian to English. The manuscript appears here in its original form with only minor editorial corrections.Publication of this historical document would not have been possible without the assistance and effort of many people. In particular, the significant help of C. de Boor, Ya.I. Fet, V.L. Kantorovich, V.N. Konovalov, and S.S. Kutateladze is acknowledged and greatly appreciated.Extended abstract55Translated from Russian by K. Kopotun.The current manuscript is devoted to the investigation of general semi-ordered spaces that are not necessarily linear. Hence, it may be considered a generalization of the work of L.V. Kantorovich [Linear semi-ordered spaces, Mat. Sbornik, 2 (1) 1937, 121–168].We say that a set Y={y} is a semi-ordered space if its elements are partially ordered using a relation “<” so that I.If y1<y2, y2<y3, then y1<y3.II.For any pair y1, y2, there exist elements y3,y4 such that y3⩽y1, y3⩽y2, y1⩽y4, and y2⩽y4.III.Every set E⊂Y bounded above has a least upper bound (supE).IV.For every set E⊂Y, there exists a countable subset E′ that has the same least upper and greatest lower bound as E. The above assumptions allow us to introduce notions of a limit superior, limit inferior, and of a convergent sequence in Y. For example, define lim¯yn=infn(sup(yn,yn+1,…)). It is possible to introduce, e.g., the limit superior differently, for example, by defining lim¯∗yn to be the least element y having the property that, for any subsequence {ynk}, there exists a subsequence {ynki} such that y⩾lim¯i→∞ynki. This type of convergence, ∗-convergence, turns out to be identical with the topological convergence that we arrive at if we turn Y into a topological space using the convergence defined initially. Relationships among various limits which we can define using the above approaches as well as some properties of these limits are studied in § 1 and § 2. In § 3, we study semi-ordered spaces equipped with a nonnegative metric function ρ(y1,y2) defined for all pairs y1, y2 such that y1⩽y2, and satisfying 1∘.ρ(y1,y2)=0 is equivalent to y1=y2.2∘.ρ(y1,y3)⩽ρ(y1,y2)+ρ(y2,y3) (y1⩽y2⩽y3).3∘.ρ(sup(y,y1),sup(y,y2))⩽ρ(y1,y2) (an analogous inequality holds with inf).4∘.If yn→y monotonically, then ρ(yn,y)→0 (or ρ(y,yn)→0).5∘.If yn monotonically tends to infinity, then the condition limn,m→∞ρ(yn,ym)=0 should not hold.Let ρ(y1,y2,…,yn)=ρ(inf(y1,…,yn),sup(y1,…,yn)). Then yn→y turns out to be equivalent to ρ(y,yn,…,yn+p)→0 when n→∞, and yn→y(∗) is equivalent to ρ(y,yn)→0. In addition, Cauchy’s convergence principle holds. Moreover, if Y is distributive, i.e., inf(y,sup(y1,y2))=sup(inf(y,y1),inf(y,y2)), then it is also strongly distributive: inf(y,supnyn)=supn(inf(y,yn)). In § 4, we study similar spaces under weaker assumptions. Particular examples of such spaces are the Hausdorff space of closed sets (see Hausdorff “Set theory”, p. 165) and the space of semicontinuous functions. § 5 is devoted to applications of the general theorems to the theory of semicontinuous functions y=f(x) that map a metric space {x}=X into a semi-ordered space {y}=Y. Under some additional assumptions (Y is regular, distributive, and between any two elements y1 and y2 such that y1<y2 there is a third element y3, y1<y3<y2) it is possible to develop a complete theory of semicontinuous functions including a theorem that every semicontinuous function is a limit of a monotone sequence of continuous functions as well a theorem on separation by a continuous function

Signatures of Radiation Reaction in Ultra-Intense Laser Fields

We discuss radiation reaction effects on charges propagating in ultra-intense laser fields. Our analysis is based on an analytic solution of the Landau-Lifshitz equation. We suggest to measure radiation reaction in terms of a symmetry breaking parameter associated with the violation of null translation invariance in the direction opposite to the laser beam. As the Landau-Lifshitz equation is nonlinear the energy transfer within the pulse is rather sensitive to initial conditions. This is elucidated by comparing colliding and fixed target modes in electron laser collisions.Comment: 8 pages, 6 figure

Persistent correlation of constrained colloidal motion

We have investigated the motion of a single optically trapped colloidal particle close to a limiting wall at time scales where the inertia of the surrounding fluid plays a significant role. The velocity autocorrelation function exhibits a complex interplay due to the momentum relaxation of the particle, the vortex diffusion in the fluid, the obstruction of flow close to the interface, and the harmonic restoring forces due to the optical trap. We show that already a weak trapping force has a significant impact on the velocity autocorrelation function C(t)= at times where the hydrodynamic memory leads to an algebraic decay. The long-time behavior for the motion parallel and perpendicular to the wall is derived analytically and compared to numerical results. Then, we discuss the power spectral densities of the displacement and provide simple interpolation formulas. The theoretical predictions are finally compared to recent experimental observations.Comment: 12 pages, 6 figure

Lorentzian regularization and the problem of point-like particles in general relativity

The two purposes of the paper are (1) to present a regularization of the self-field of point-like particles, based on Hadamard's concept of ``partie finie'', that permits in principle to maintain the Lorentz covariance of a relativistic field theory, (2) to use this regularization for defining a model of stress-energy tensor that describes point-particles in post-Newtonian expansions (e.g. 3PN) of general relativity. We consider specifically the case of a system of two point-particles. We first perform a Lorentz transformation of the system's variables which carries one of the particles to its rest frame, next implement the Hadamard regularization within that frame, and finally come back to the original variables with the help of the inverse Lorentz transformation. The Lorentzian regularization is defined in this way up to any order in the relativistic parameter 1/c^2. Following a previous work of ours, we then construct the delta-pseudo-functions associated with this regularization. Using an action principle, we derive the stress-energy tensor, made of delta-pseudo-functions, of point-like particles. The equations of motion take the same form as the geodesic equations of test particles on a fixed background, but the role of the background is now played by the regularized metric.Comment: 34 pages, to appear in J. Math. Phy

Magnetic mirror cavities as THz radiation sources and a means of quantifying radiation friction

We propose a radiation source based on a magnetic mirror cavity. Relativistic electrons are simulated entering the cavity and their trajectories and resulting emission spectra are calculated. The uniformity of the particle orbits is found to result in a frequency comb in terahertz range, the precise energies of which are tuneable by varying the electron's $\gamma$-factor. For very high energy particles radiation friction causes the spectral harmonics to broaden and we suggest this as a possible way to verify competing classical equations of motion.Comment: 8 pages, 10 figure

A kinetic model of radiating electrons

A kinetic theory is developed to describe radiating electrons whose motion is governed by the Lorentz-Dirac equation. This gives rise to a generalized Vlasov equation coupled to an equation for the evolution of the physical submanifold of phase space. The pathological solutions of the 1-particle theory may be removed by expanding the latter equation in powers of τ ≔ q 2/6πm. The radiation-induced change in entropy is explored and its physical origin is discussed. As a simple demonstration of the theory, the radiative damping rate of longitudinal plasma waves is calculated

New Environmental Demands and the Future of the Helsinki−Tallinn Freight Route

The environmental friendliness of short sea shipping has been justified in Europe by the ensuing lower congestion at hinterlands and unneeded large-scale infrastructure investments on roads and railways. However, the attractiveness of short sea shipping is about to change. This is because of increasing environmental regulations (International Maritime Organization (IMO) sulfur regulation in the Baltic Sea and planned CO2 emissions trading) and increased world market oil prices. In this research, we analyze this potential change using data envelopment analysis on the existing transportation chain alternatives in the Helsinki (Finland)−Tallinn (Estonia) short sea route (chains using either roro, ropax or container ships). The analysis also includes the planned railway tunnel between the two cities. On the basis of our findings, the current truck and semi-trailer-based transportation is challenged by containers, irrespective of how they are carried (ship type). In the long term, for reasons of emissions and oil independency, the possibility of tunnel construction would make it vital to have container ship operations available along this route. The forthcoming change is not radical, but rather evolutionary and long term oriented

Single thyroid nodules

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