Physics

"More than a third part of a century ago, in the library of an ancient town, a youth might have been seen tasting the sweets of knowledge to see how he liked them. He was of somewhat unprepossessing appearance, carrying on his brow the heavy scowl that the "mostly-fools" consider to mark a scoundrel. In his father's house were not many books, so it was like a journey into strange lands to go book-tasting. Some books were poison; theology and metaphysics in particular they were shut up with a bang. But scientific works were better; there was some sense in seeking the laws of God by observation and experiment, and by reasoning founded thereon. Some very big books bearing stupendous names, such as Newton, Laplace, and so on, attracted his attention. On examination, he concluded that he could understand them if he tried, though the limited capacity of his head made their study undesirable. But what was Quaternions? An extraordinary name! Three books; two very big volumes called Elements, and a smaller fat one called Lectures. What could quaternions be? He took those books home and tried to find out. He succeeded after some trouble, but found some of the properties of vectors professedly proved were wholly incomprehensible. How could the square of a vector be negative? And Hamilton was so positive about it. After the deepest research, the youth gave it up, and returned the books. He then died, and was never seen again. He had begun the study of Quaternions too soon."

"My own introduction to quaternionics took place in quite a different manner. Maxwell exhibited his main results in quaternionic form in his treatise. I went to Prof Tait's treatise to get information, and to learn how to work them. I had the same difficulties as the deceased youth, but by skipping them, was able to see that quaternionics could be employed consistently in vectorial work. But on proceeding to apply quaternionics to the development of electrical theory, I found it very inconvenient. Quaternionics was in its vectorial aspects antiphysical and unnatural, and did not harmonise with common scalar mathematics. So I dropped out the quaternion altogether, and kept to pure scalar and vectors, using a very simple vectorial algebra in my papers from 1883 onward. The paper at the beginning of vol. 2 of my Electrical Papers may be taken as a developed specimen; the earlier work is principally concerned with the vector differentiator ∇ and its applications, and physical interpretations of the various operations. Up to 1888 I imagined that I was the only one doing vectorial work on positive physical principles; but then I received a copy of Prof. Gibbs's Vector Analysis (unpublished, 1881-4)."

"Frobenius' Theorem. Over the real number field there exist precisely three associative s, namely the real numbers, the complex numbers, and the real quaternions."

"It is a curious fact in the history of mathematics that discoveries of the greatest importance were made simultaneously by different men of genius. The classical example is the... development of the infinitesimal calculus by Newton and Leibniz. Another case is the development of vector calculus in Grassmann's Ausdehnungslehre and Hamilton's Calculus of Quaternions. In the same way we find analytic geometry simultaneously developed by Fermat and Descartes."

"[Q]uaternions form the appropriate algebraic basis for a description of nature whenever we have to deal either with pseudoreal group representations or with co-representations of Wigner's Type II. The context in which quaternions arose historically, in a study of the three-dimensional rotation group, can now be seen to be an extremely special case of this general principle. Every group which admits pseudoreal representations equally admits a natural description in terms of real quaternions."

"The history of geometry may be conveniently divided into five periods. The first extends from the origin of the science to about A. D. 550, followed by a period of about 1,000 years during which it made no advance, and in Europe was enshrouded in the darkness of the middle ages; the second began about 1550, with the revival of the ancient geometry; the third in the first half of the 17th century, with the invention by Descartes of analytical or modern geometry; the fourth in 1684, with the invention of the differential calculus; the fifth with the invention of by Monge in 1795. The quaternions of Sir William Rowan Hamilton, the Ausdehnungslehre of Dr. Hermann Grassmann, and various other publications, indicate the dawn of a new period. Whether they are destined to remain merely monuments of the ingenuity and acuteness of their authors, or are to become mighty instruments in the investigation of old and the discovery of new truths, it is perhaps impossible to predict."

"Mr. McAulay asks: "What is the first duty of the physical vector analyst quâ physical vector analyst?" The answer is... to present the subject in such a form as to be most easily acquired, and most useful when acquired. ...What then is the cause of the fact ...all of us deplore? ...We need only a glance at the volumes in which Hamilton set forth his method. No wonder that physicists and others failed to perceive the possibilities of simplicity, perspicuity, and brevity... in a system presented... in ponderous volumes of 800 pages. ...[I]f we turn to his earlier papers on Quaternions in the Philosophical Magazine... we find... "On Quaternions; or on a New System of Imaginaries in Algebra," and in them we find a great deal about imaginaries and very little of a vector analysis. To show how slowly the system of vector analysis developed itself in the quaternionic nidus, we need only say that the symbols S, V, and ∇ do not appear until two or three years after the discovery of quaternions. In short it seems to have been only a secondary object with Hamilton to express the geometrical relations of vectors... it was never allowed to give shape to his work. ...[I]s it not discouraging to be told that in order to use the quaternionic method one must give up the progress which he has already made in the pursuit of his favourite science and go back to the beginning and start anew on a parallel course? ...Whatever is special, accidental, and individual, will die, as it should; but that which is universal and essential should remain as an organic part of the whole intellectual acquisition. If that which is essential dies with the accidental, it must be because the accidental has been given the prominence which belongs to the essential. ...In Italy they say all roads lead to Rome. In mechanics, , astronomy, physics, all study leads to the consideration of certain relations and operations. These are the capital notions; these should have the leading parts in any analysis suited to the subject."

"Examples of geometric phases abound in many areas of physics. Many familiar problems that we do not ordinary associate with geometric phases may be phrased in terms of them. Often, the result is a clearer understanding of the structure of the problem, and of its solution."

"The geometric phase acquired by the eigenstates of cycled quantum systems is given by the flux of a two-form through a surface in the system’s parameter space. We obtain the classical limit of this two-form in a form applicable to systems whose classical dynamics is chaotic. For integrable systems the expression is equivalent to the Hannay two-form. We discuss various properties of the classical two-form, derive semiclassical corrections to it (associated with classical periodic orbits), and consider implications for the semiclassical density of degeneracies."

"Here as he walked by, on the 16th of October 1843 Sir William Rowan Hamilton in a flash of genius discovered the fundamental formula for quaternion multiplication i^2 = j^2 = k^2 = ijk = -1 & cut it on a stone of this bridge."

"Quantum theory may be formulated using s over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. ...[P]roblems can be resolved if we treat real, complex and quaternionic quantum theory as part of a unified structure. Dyson called this structure the "three-fold way". ... This three-fold classification sheds light on the physics of time reversal symmetry, and it already plays an important role in particle physics. ...There are precisely four 'normed division algebras': the real numbers \mathbb{R}, the complex numbers \mathbb{C}, the quaternions \mathbb{H} and the octonions \mathbb{O}. Roughly speaking, these are the number systems extending the reals that have an ‘absolute value’ obeying the equation |xy| = |x| |y|. Since the octonions are nonassociative [their use] proves difficult... except in a few special cases. ...[I]nstead of being distinct alternatives, real, complex and quaternionic quantum mechanics are three aspects of a single unified structure."

"Whenever a quantum system undergoes a cyclic evolution governed by a slow change of parameters, it acquires a phase factor: the geometric phase. Its most common formulations are known as the Aharonov–Bohm phase and the Pancharatnam and Berry phase, but both earlier and later manifestations exist. Although traditionally attributed to the foundations of quantum mechanics, the geometric phase has been generalized and become increasingly influential in many areas from condensed-matter physics and optics to high-energy and particle physics and from fluid mechanics to gravity and cosmology. Interestingly, the geometric phase also offers unique opportunities for quantum information and computation. In this Review, we first introduce the Aharonov–Bohm effect as an important realization of the geometric phase. Then, we discuss in detail the broader meaning, consequences and realizations of the geometric phase, emphasizing the most important mathematical methods and experimental techniques used in the study of the geometric phase, in particular those related to recent works in optics and condensed-matter physics."

"The real numbers are the dependable breadwinner of the family, the complete ordered field we all rely on. The complex numbers are a slightly flashier but still respectable younger brother: not ordered, but algebraically complete. The quaternions, being noncommutative, are the eccentric cousin who is shunned at important family gatherings. But the octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative."

"One of the simplest chemical exchange reactions involves a system of three hydrogen atoms: H+H2→H2+H. Surely, chemists have felt, one should be able to calculate the cross sections for this reaction from first principles. But the computations have not been easy. Only in the last six years or so have theorists, aided by efficient methodologies and access to supercomputers, been able to predict the cross sections in sufficient detail for comparison with experiments, which themselves have evolved in precision. The agreement has been good—well, almost. Small discrepancies, especially at higher total energies, stubbornly refused to yield to adjustments in either the calculations or the experiments. Now Yi‐Shuen Mark Wu and Aron Kuppermann of Caltech have erased these pesky discrepancies by including a topological effect known as the geometric phase. Michael Berry (University of Bristol) has called attention to the presence of this phase, which now bears his name, in a wide variety of physical systems."

"If I wished to attract the student of any of these sciences to an algebra for vectors, I should tell him that the fundamental notions of this algebra were exactly those with which he was daily conversant. ...I should call his attention to the fact that Lagrange and Gauss used the notation (αβγ) to denote precisely the same as Hamilton by his S(αβγ) except that Lagrange limited the expression to s, and Gauss to vectors of which the length is the secant of the latitude, and I should show him that we have only to give up these limitations, and the expression (in connection with the notion of geometrical addition) is endowed with an immense wealth of transformations. I should call his attention to the fact that the notation [r_1r_2], universal in the theory of orbits, is identical with Hamilton's V(\rho_1\rho_2) except that Hamilton takes the area as a vector... I confess that one of my objects was to show that a system of vector analysis does not require any support from the notion of the quaternion, or... of the imaginary in algebra."

"The quantum mechanics of two identical particles with spin S in three dimensions is reformulated by employing not the usual fixed spin basis but a transported spin basis that exchanges the spins along with the positions. Such a basis, required to be smooth and parallel-transported, can be generated by an ‘exchange rotation’ operator resembling angular momentum. This is constructed from the four harmonic oscillators from which the two spins are made according to Schwinger's scheme. It emerges automatically that the phase factor accompanying spin exchange with the transported basis is just the Pauli sign, that is (−1)2S. Singlevaluedness of the total wavefunction, involving the transported basis, then implies the correct relation between spin and statistics. The Pauli sign is a geometric phase factor of topological origin, associated with non-contractible circuits in the doubly connected (and non-orientable) configuration space of relative positions with identified antipodes. The theory extends to more than two particles."

"One of the reasons for being interested in the geometric phase is that it connects a number of different areas."

"Closely akin to his third and fourth propositions is Riemann's fifth proposition, that continuous quantities are coördinate with discrete quantities, both being in their nature multiples or aggregates, and therefore species of the same genus. This pernicious fallacy is one of the traditional errors current among mathematicians, and has been prolific of innumerable delusions. It is this error which has stood in the way of the formation of a rational, intelligible, and consistent theory of irrational and imaginary quantities, so called, and has shrouded the true principles of the doctrine of "complex numbers" and of the calculus of quaternions in an impenetrable haze."

"Prof. Tait has spoken of the calculus of quaternions as throwing off in the course of years its early Cartesian trammels. I wonder that he does not see how well the progress in which he has led may be described as throwing off the yoke of the quaternion. A characteristic example is seen in the use of the symbol ∇. Hamilton applies this to a vector to form a quaternion, Tait to form a linear vector function. ...Now I appreciate and admire the generous loyalty toward one whom he regards as his master which has always led Prof. Tait to minimise the originality of his own work in regard to quaternions and write as if everything was contained in the ideas which flashed into the mind of Hamilton at the classic . But... we owe duties to our scholars as well as to our teachers, and the world is too large, and the current of modern thought is too broad, to be confined by the ipse dixit [he says] even of a Hamilton"

"Hari Seldon devised psychohistory by modeling it upon the kinetic theory of gases. Each atom or molecule in a gas moves randomly so that we can't know the position or velocity of any one of them. Nevertheless, using statistics, we can work out the rules governing their overall behavior with great precision. In the same way, Seldon intended to work out the overall behavior of human societies even though the solutions would not apply to the behavior of the individual human beings."

"In this book I have tried... to make clearly comprehensible the path-breaking works of Clausius and Maxwell. The reader may not think badly of me for finding also a place for my own contributions. These were cited respectfully in Kirchhoff's lectures [on Maxwell's kinetic theory] and in Poincare’s Thermodynamique at the end, but were not utilized where they would have been relevant. From this I concluded that a brief presentation, as easily understood as possible, of some of the principal results of my efforts might not be superfluous. Of great influence on the content and presentation was what I have learned at the unforgettable meeting of the British Association in Oxford and the subsequent letters of numerous English scientists, some private and some published in Nature. I intend to follow Part I by a second part, where I will treat the van der Waals theory, gases with polyatomic molecules, and dissociation. ...Unfortunately it was often impossible to avoid the use of long formulas to express complicated trains of thought, and... to many who do not read over the whole work, the results will perhaps not seem to justify the effort expended. Aside from many results of pure mathematics which, though likewise apparently fruitless at first, later become useful in practical science as soon as our mental horizon has been broadened, even the complicated formulas of Maxwell’s theory of electromagnetism were often considered useless before Hertz’s experiments. I hope this will not also be the general opinion concerning gas theory!"

"Boltzmann decided to publish his lectures, in which the most important parts of the theory, including his ...contributions, were carefully explained. ...[H]e included his mature reflections and speculations on such questions as the nature of irreversibility and the justification for using statistical methods in physics. His Vorlesungen über Gastheorie was... the standard reference... for advanced researchers, ...[and] a popular textbook ...for the first quarter of the [20th] century ...The reason why the classical theory works is that, while the internal structure of molecules must be described by quantum mechanics, the interaction between two molecules can be fairly well described by a classical model which ignores this structure and simply uses a postulated force law whose parameters can be chosen to fit experimental data. ...Aside from phenomena at very high densities or very low temperatures, the only property that the classical theory fails to account for is the ratio of specific heats. ...Boltzmann ...simply concludes that for some unknown reason all the possible internal motions of a molecule do not have an equal share in the total energy, and takes this into account as an empirical fact."

"Boyle... proposed a theoretical explanation for the elasticity of air... "a heap of little bodies, lying upon one another"... The atoms are said to behave like springs... Boyle also tried the "crucial experiment" which was to help overthrow his own theory in favor of the kinetic theory two centuries later, though he did not realize its significance... Experiment No. 26... places a pendulum in the evacuated chamber... [A]bsence of air makes hardly any difference to the period of the swings or the time... to come to rest. In 1859, James Clerk Maxwell deduced from the kinetic theory that the viscosity of a gas should be independent of its density... which would be very hard to explain on the basis of Boyle's theory. ...Neither Boyle nor Newton claimed that the hypothesis of repulsive forces between atoms is the only correct explanation for gas pressure; both were willing to leave the question open. Boyle mentions the Descartes theory of vortices (1644)... somewhat closer in spirit to the kinetic theory since it relies more heavily on the rapid motion of the parts of the atom as a cause of repulsion. (Though Descartes did not believe in "atoms" in the classical sense.) Nevertheless, the Boyle-Newton theory of gases was apparently accepted by most scientists until about the middle of the 19th century, when the kinetic theory finally managed to overcome Newton's authority."

"It is difficult to understand the relative lack of progress in gas theory during the 18th century ...[T]here was little interest in the properties of freely moving atoms. The atoms in gas were... conceived as... suspended in the ether, although they could vibrate or rotate enough to keep other atoms from coming too close. This model was... awkward... mathematically, as... seen from an... attempt by Leonhard Euler in 1727. ...[O]ne contribution from this period has been... recognized as the first kinetic theory of gases. This is Daniel Bernoulli's derivation of the gas laws from a "billiard ball" model—in 1738... [H]is kinetic theory is... a small part of a treatise [Hydrodynamica (1738)] on hydrodynamics... Bernoulli's formulation and... applications of the principle of conservation of mechanical energy (...' ..."living force" ...) were ...more important than the fact that he proposed a kinetic theory ...a century ahead of its time ...Heat was still regarded as a substance ...Bernoulli's assumption that heat was nothing but atomic motion was unacceptable, especially to scientists interested in... radiant heat. The assumption that atoms could move freely through space until they collided like billiard balls... neglected the drag of the ether and oversimplified the interaction between atoms. ...When physics reached the stage of development at which the kinetic theory no longer conflicted with established principles, ...[it] had almost been forgotten and had to be rediscovered. ...In a very real sense, the man who persuades the world to adopt a new idea has accomplished as much as the man who conceived that idea."

"The kinetic theory of gases is a small branch of physics which has passed from the stage of excitement and novelty into staid maturity. ...Formerly it was hoped that the subject of gases would ultimately merge into a general kinetic theory of matter; but the theory of condensed phases... today, involves an elaborate and technical use of wave mechanics, and for this reason it is best treated as a subject in itself. The scope of the present book is, therefore, the traditional kinetic theory of gases. ...[A]n account has been included of the wave-mechanical theory, and especially of the degenerate Fermi-Dirac case... There is also a concise chapter on , which... may be of use as an introduction... [T]he discussion of electrical phenomena has been abbreviated... the latter voluminous subject is best treated separately. ...[F]undamental parts have been explained... [as] to be within the reach of college juniors and seniors. The... wave mechanics and statistical mechanics... are of graduate grade. ...[A] number of carefully worded theorems have been inserted in the guise of problems, without proof... to give... a chance to apply... lines of attack exemplified in the text. To facilitate use as a reference book, definitions have been repeated freely, I hope not ad nauseam. ...Ideas have been drawn freely from ...books such as ...of Jeans and Loeb..."

"The last thirty years have seen the beginning and development of a new period in physics and chemistry, namely the atomic period. In contrast to the period preceding it where nature's processes were described in terms of continua, recent developments have emphasized the discrete structure of the submicroscopic universe. Thus, today one hears of the atoms of matter, the atoms of electricity, and even the atoms of energy, the quanta. ...[T]he atomic theory of matter is the oldest and perhaps the most complete. ...[B]ecause of its relative simplicity the problem of the atomic theory of gases, in the form of the kinetic theory of gases, has attained the highest degree of perfection in this field. Its admirable methods of analysis are therefore indispensable... This book... endeavors to develop the various concepts... independently...Besides a simple introduction of each concept it gives derivations... elementary ones, using little or no calculus; more advanced classical derivations; and in some cases the most recent developments available. It also contains the comparison of the theoretical deductions with modern experiment and a critique of the theories."

"The science of Thermodynamics, founded by the labors of these three illustrious men [Nicolas Léonard Sadi Carnot, William Thomson & Rudolf Clausius], has led to the most important developments in all departments of physical science. It has pointed out relations among the properties of bodies which could scarcely have been anticipated in any other way; it has laid the foundation for the Science of Chemical Physics; and, taken in connection with the , as developed by Maxwell and Boltzmann, it has furnished a general view of the operations of the universe which is far in advance of any that could have been reached by purely dynamical reasoning."

"The influence of Quetelet's ideas spread throughout the sciences, even to the physical sciences. The two primary founders of the modern kinetic theory of gases, based on considerations of probability, were James Clerk Maxwell and Ludwig Boltzmann. Both acknowledged their debt to Quetelet. ...[H]istorians generally consider the influence of the natural sciences on the social sciences, whereas in the case of Maxwell and Boltzmann, there is an influence of the social sciences on the natural sciences, as Theodore Porter has shown."

"The old mechanical and atomic hypotheses have, during recent years, become so plausible that they have ceased to seem like hypotheses; atoms are no longer just a convenient fiction. It seems almost as if we could see them, now that we know how to count them. ...The kinetic theory of gases has thus received unexpected corroboration. ...The remarkable counting of the number of atoms by Perrin completed the triumph of the atomic theory. ...In the processes used with the Brownian phenomenon, or in those used for the law of radiation, we do not deal directly with the number of atoms, but with their degrees of freedom of movement. In that process where we consider the blue of the sky, the mechanical properties of the atoms come into play; the atoms are looked upon as producing an optical discontinuity. ...The atom of the chemist is now a reality. But that does not mean that we have reached the ultimate limit of the divisibility of matter. When Democritus invented the atom he considered it as the absolutely indivisible element within which there would be nothing further to distinguish. That is what the word meant in Greek. ... the atom of the chemist would not have satisfied him since that is not indivisible; it is not a true element; it is not free from mystery, from secrets. The chemist's atom is a universe. Democritus would have considered, even after so much trouble in finding it, that we were still only at the beginning of our search—these philosophers are never satisfied. ...This atom disintegrates into yet smaller atoms. What we call is the perpetual breaking up of atoms. ...Each atom is like a sort of solar system where the small negative electrons play the role of planets revolving around the great... sun. ...the atom of a radioactive body is a universe within itself and a world subject to chance."

"The idea of a Kinetic Theory of Gases originated with J. Bernouilli about the middle of the last century, but the first establishment of the theory on a scientific basis is due to Professor Clausius. During the last few years the theory has been greatly developed by many physicists, especially by Professor Clerk Maxwell in England and Professor Clausius and Dr. Ludwig Boltzmann... and although still beset by formidable difficulties, it has succeeded in explaining most of the established laws of gases in so remarkable a manner as to render it well worthy of the attentive consideration of scientific men. ...For the most part I have followed the method of treatment adopted by Dr. Ludwig Boltzmann in some very interesting memoirs ..."

"If we study the history of science we see happen two inverse phenomena, so to speak. Sometimes simplicity hides under complex appearances; sometimes it is the simplicity which is apparent, and which disguises extremely complicated realities. ...What is more complicated than the confused movements of the planets? What simpler than Newton's law? ...In the kinetic theory of gases, one deals with molecules moving with great velocities, whose paths, altered by incessant collisions, have the most capricious forms... The observable result is Mariotte's simple law. ...The law of great numbers has reestablished simplicity in the average. ...No doubt, if our means of investigation should become more and more penetrating, we should discover the simple under the complex, then the complex under the simple, then again the simple under the complex, and so on, without our being able to foresee what will be the last term. We must stop somewhere, and that science may be possible, we must stop when we have found simplicity. This is the only ground on which we can rear the edifice of our generalizations."

"The researches of Galileo, followed up by Huygens and others, led to those modern conceptions of Force and Law, which have revolutionized the intellectual world. The great attention given to mechanics in the seventeenth century soon so emphasized these conceptions as to give rise to the Mechanical Philosophy, a doctrine that all the phenomena of the physical universe are to be explained upon mechanical principles. Newton's great discovery imparted a new impetus to this tendency. The old notion that heat consists in an agitation of corpuscles was now applied as an explanation to the chief properties of gases. The first suggestion in this direction was that the pressure of gases is explained by the battering of the particles against the walls of the containing vessel, which explained Boyle's law of the compressibility of air. Later, the expansion of gases, Avogadro's chemical law, the diffusion and viscosity of gases, and the action of Crooke's radiometer were shown to be consequences of the same kinetical theory; but other phenomena, such as the ratio of the specific heat at constant volume to that at constant pressure, require additional hypotheses, which we have little reason to suppose are simple, so that we find ourselves quite afloat. In like manner with regard to light..."

"The first constant... is connected with the definition of temperature. If temperature were defined as the mean of a molecule in a , which is a minute energy indeed, this constant would have the value ⅔. But in the conventional scale of temperature the constant ...[instead] assumes an extremely small value... intimately connected with the energy of a single molecule... [I]ts accurate determination would lead to the calculation of the mass of a molecule and... associated magnitudes. This constant is frequently termed Boltzmann's constant, although to the best of my knowledge Boltzmann... never introduced it (...he, as appears from... his statements, never believed it would be possible to determine this constant accurately)..."

"The first edition of this book appeared in 1877, at the time of the most rapid and beautiful development of the kinetic theory of gases. About twenty years before, the founders... Kronig and Clausius, had explained the expansive tendency of gases, and had calculated their pressure on the assumption that the smallest particles of gases do not repel each other, but are in rapid motion. From the theory based on this supposition not only were the laws of gases... deduced... but also new laws, hitherto undreamt of, were discovered... [and] afterwards confirmed... by experiment. These results, which we owe to Maxwell and Clausius, quickly won to the theory many friends and adherents. ...I undertook ...to exhibit the ...theory ...such ...as to be more easily intelligible ...especially to chemists and other natural philosophers to whom mathematics are not congenial. ...I endeavoured ...not only to develop the theory by calculation, but ...to support it by observation and found it on experiment. I... collected... and summarised, the observations by which the admissibility of the theory might be tested and its correctness proved. ...The mathematical discussions form ...an Appendix which ...need not be studied by every reader ..."

"Now, although the plans of the edifice of the electromagnetic theory of light were laid in 1880 by H. A. Lorentz, and even indicated much earlier by W. Weber, a full 10 years were required before the discoveries of Heinrich Hertz gave the impetus to collect the building stones and work them into shape. In the years 1890-93 a number of works appeared by F. Richarz, H. Ebert and G. Johnstone Stoney, mostly dealing with the mechanism of the emission of luminous vapours, and in which attempts are made, on the basis of the kinetic theory of gases, to determine the magnitude of the elementary electrical quantity, called by Stoney by the now universally accepted name of electron. ...H. Ebert proved that the amplitude of an electron in luminous sodium vapour need only be a small fraction of a molecular diameter in order to excite a radiation of the absolute intensity determined by E. Wiedemann. The way of determining the amount of electricity contained in the electron is very simple. The quantity of electricity required for the electrolytic evolution of 1 cubic cm. of any monatomic gas is divided by Loschmidt's number—i.e., the number of gas molecules contained in 1 cubic cm."

"It will be shown in this paper that, according to the molecular-kinetic theory of heat, bodies of microscopically visible size suspended in liquids must, as a result of thermal molecular motions, perform motions of such magnitude that these motions can easily be detected by a microscope. It is possible that the motions to be discussed here are identical with the so-called "Brownian molecular motion"; however, the data available to me on the latter are so imprecise that I could not form a definite opinion on this matter. If it is really possible to observe the motion to be discussed here, along with the laws it is expected to obey, then classical thermodynamics can no longer be viewed as strictly valid even for microscopically distinguishable spaces, and an exact determination of the real size of atoms becomes possible. Conversely, if the prediction of this motion were to be proved wrong, this fact would provide a weighty argument against the molecular-kinetic conception of heat."

"In 1909 Perrin suspended particles of in a liquid of slightly lower density, and found that the heavy particles did not sink to the bottom of the lighter liquid; they were prevented from doing so by their own Brownian movements. If the liquid had been infinitely fine-grained, with molecules of infinitesimal size and weight, every solid particle would have had as many impacts from above as below; these impacts, coming in a continuous stream, would have just cancelled one another out, so that each particle would have been free to fall to the bottom under its own weight. But when they were bombarded by molecules of finite size and weight, the solid particles were hit, now in one direction and now in another, and so could not lie inertly on the bottom of the vessel. From the extent to which they failed to do this, Perrin was able to form an estimate of the weights of the molecules of the liquid... and this agreed so well with other estimates that there could be but little doubt felt as to the truth either of the kinetic theory of liquids, or of the associated explanation of the Brownian movements."

"One of the most important and interesting aims of is to explain the properties of matter in terms of the motions and spatial arrangements of atoms and molecules. This aim has been more nearly achieved in the physical chemical study of gases at low pressures [below a few atmospheres at ordinary temperatures] than the study of matter in any other conditions. ... The structure of gases at these pressures is particularly simple: such gases are collections of molecules which move randomly in space and which collide with each other relatively infrequently—that is, the molecules are so far apart that much of the time they exert little influence on each other. ...[T]he properties of the gaseous state play a role in many important practical processes, such as [in]... the internal combustion engine, the function of the lungs, the motions of the winds across the earth and the flight of airplanes. Gases... provide a useful and pedagogically attractive starting point for the introduction of students to physical chemistry."

"... In the case of the Carrington event of 1859, the most severe coronal mass ejection known to have occurred, the propagation time between the Sun and the Earth, at a speed of 2,300 kilometres per second, was seventeen and a half hours. The way to avert the most serious impacts would be to make adjustments to the operation of the electricity grids before the storm struck (see Space Studies Board 2008, Chapter 7). The necessary actions would have to be taken very quickly and in a coordinated way in order to be effective, so they would have to be carefully planned in advance, preferably in an international context."

"... It was impossible, on first witnessing an appearance so similar to a sudden conflagration, not to expect a considerable result in the way of alteration of the details of the group in which it occurred; and I was certainly surprised, on referring to the sketch which I had carefully and satisfactorliy (and I may add fortunately) finished before the ocurrence, at finding myself unable to recognise any change whatever as having taken place. The impression left upon me is, that the phenomenon took place at an elevation considerably above and over the great group in which it was seen projected."

"Without warning, two beads of searing white light, bright as forked lightning but rounded rather than jagged and persistent instead of fleeting, appeared over the monstrous sunspot group. Momentarily taken by surprise, Carrington assumed that a ray of sunlight had found its way through the shadow-screen attached to the telescope. He reached out and jiggled the instrument, expecting the errant ray to zip wildly across the image. Instead, it stayed doggedly fixed in its position on the sunspot group. Whatever it was, it was not some stray reflection; it was coming from the Sun itself. As he stared, dumfounded, the two spots of light intensified and became kidney shaped."

"In this paper it is shown that a star must experience dynamical friction, i.e., it must suffer from a systematic tendency to be decelerated in the direction of its motion. This dynamical friction which stars experience is one of the direct consequences of the fluctuating force acting on a star due to the varying complexion of the near neighbors. From considerations of a very general nature it is concluded that the coefficient of dynamical friction, \eta, must be of the order of the reciprocal of the time of relaxation of the system. Further, an independent discussion based on the two-body approximation for stellar encounters leads to the following explicit formula for the coefficient of dynamical friction: \eta = 4\pi m_1 (m_1 + m_2)G^2/v^3 log_e [D_0\overline {|u|^2}/G(m_1+m_2)] \int_{0}^{v} N(v_1) \,dv_1, where m_l and m_2 denote the masses of the field star and the star under consideration, respectively; G, the constant of gravitation; D_0 the average distance between the stars; \overline {|u|^2}, the mean square velocity of the stars; N(v_1) dv_1, the number of field stars with velocities between v_1 and v_1 + dv_1; and, finally, v, the velocity of the star under consideration. It is shown that the foregoing formula for η is in agreement with the conclusions reached on the basis of the general considerations. Finally, some remarks are made concerning the further development of these ideas on the basis of a proper statistical theory."

"We investigate dynamical friction on a test object (such as a bar or satellite) which rotates or revolves through a spherical stellar system. We find that frictional effects arise entirely from near-resonant stars and we derive an analog to Chandrasekhar's dynamical friction formula which applies to spherical systems. We show that a formula of this type is valid so long as the angular speed of the test object changes sufficiently rapidly. If the angular speed is slowly changing two new effects appear: a reversible dynamical feedback which can stabilize or destabilize the rotation speed, and permanent capture of near-resonant stars into librating orbits. We discuss orbital decay of satellites in the light of these results."

"A test particle traveling through a collisionless gravitating background suffers a dissipative drag force known as dynamical friction. As with other dissipative forces, this friction must be related to fluctuations in the underlying medium (fluctuation-dissipation theorem). However, this long recognized aspect of the force did not easily yield to analysis until now, and Chandrasekhar’s celebrated formula was obtained by considering momentum exchanges resulting from encounters between a test particle and field particles which were ideal- ized as occurring sequentially. In this paper we return to the underlying basic physics and develop a theory of the interaction of the test particle with the stochastic force of the background. This enables us to derive in a unified way the Chandrasekhar formula for the friction (for the full range of m/M) and the heating of the particle by background fluctuations."

"Two developments in the late 1960s and early 1970s set the stage for supergravity. First the standard model took shape and was decisively confirmed by experiments. The key theoretical concept underlying this process was gauge symmetry, the idea that symmetry transformations act independently at each point of spacetime. ... The other development was global (also called rigid) supersymmetry ... It is the unique framework that allows fields and particles of different spin to be unified in representations of an algebra system called a superalgebra."

"Supergravity is a theory of gravity which has supersymmetry, a symmetry between bosons and fermions. Supersymmetry in supergravity is a local symmetry like the gauge symmetry in the standard theory of particle physics. The gauge field of the local supersymmetry is the Rarita-Schwinger field, which represents a particle with spin called a graviton. Supersymmetry also has a local symmetry under the general coordinate transformation, whose gauge field is the gravitational field. An important role of supergravity is in its relation to superstring theory. ... Supergravity provides a low energy effective theory of the massless sector of superstring theory and can be used to study its low energy properties."

"Inflation can be caused by the potential energy of a scalar field. Such a potential must be relatively flat in order to guarantee long duration of inflation and small deviation of scale invariance of primordial density fluctuations. However, the flatness of the scalar potential can be easily destroyed by radiative corrections. One of the leading theories to protect a scalar field from radiative corrections is supersymmetry (SUSY), which also gives an attractive solution to the (similar) hierarchy problem of the standard model (SM) of particle physics as well as the unification of the three gauge couplings. In particular, its local version, supergravity, would govern the dynamics of the early Universe, when high energy physics was important. Thus, it is quite natural to consider inflation in the framework of supergravity. However, in fact, it is a non-trivial task to incorporate inflation in supergravity. This is mainly because a SUSY breaking potential term, which is indispensable to inflation, generally gives a would-be inflaton an additional mass, which spoils the flatness of an inflation potential ..."

"The question of the possibility for a completion of quantum mechanics received its most famous (partial) answer in 1964 by, again, Bell ... He proved what today is known simply as Bell's theorem, to wit, that is such a more complete description exists, it cannot be local, i.e. dependent only on the events in a system's past lightcone, and agree with quantum mechanics in all instances. To this day, this result forms the paradigm example of a 'no-go' theorem."

"... If you start off with switches and gears, or whatever, you can never construct a universe in which you see quantum mechanical phenomena, according to Bell. We call such a thing a 'no-go theorem'. You may already suspect that I still believe in the hidden variables hypothesis. Surely our world must be constructed in such an ingenious way that some of the assumptions that Einstein, Bell and others found quite natural will turn out to be wrong. But how this will come about, I do not know. Anyway, for me, the hidden variables hypothesis is still the best way to ease my conscience about quantum mechanics. And as for 'no-go theorems', we will encounter several of these and discuss their fate."

"One possibility that comes to mind is that the spin-two graviton might arise as a composite of two spin-one gauge bosons. This interesting idea would seem to be rigorously excluded by a no-go theorem of Weinberg & Witten ... The Weinberg–Witten theorem appears to assume nothing more than the existence of a Lorentz-covariant energy momentum tensor, which indeed holds in gauge theory. The theorem does forbid a wide range of possibilities, but (as with several other beautiful and powerful no-go theorems) it has at least one hidden assumption that seems so trivial as to escape notice, but which later developments show to be unnecessary. The crucial assumption here is that the graviton moves in the same spacetime as the gauge bosons of which it is made!"