Historically, Ehrenfest’s theorem is the first one which shows that classical physics can emerge from quantum physics as a kind of approximation. We recall the theorem in its original form, and we highlight its generalizations to the relativistic Dirac particle and to a particle with spin and izospin. We argue that apparent classicality of the macroscopic world can probably be explained within the framework of standard quantum mechanics.

This article highlights and discusses the Ehrenfests’ use of toy models to explore irreversibility in statistical mechanics. In particular, we explore their urn and P–Q models and highlight that, while the former was primarily used to provide a simple counter-example to Zermelo’s objection to Boltzmann’s statistical mechanical underpinning of the Second Law of Thermodynamics, the latter was intended to highlight the role and importance of the Stoßzahlansatz as a cause of the tendency of systems to exhibit entropy increase. We also (...) explain the sense in which these models are toy models and why agents can use them, as the Ehrenfests’ did, to carry out this important work, despite the fact that they do not represent any real system. (shrink)

This study attempts to spell out more explicitly than has been done previously the connection between two types of formal correspondence that arise in the study of quantum–classical relations: one the one hand, deformation quantization and the associated continuity between quantum and classical algebras of observables in the limit \, and, on the other, a certain generalization of Ehrenfest’s Theorem and the result that expectation values of position and momentum evolve approximately classically for narrow wave packet states. While deformation (...) quantization establishes a direct continuity between the abstract algebras of quantum and classical observables, the latter result makes in-eliminable reference to the quantum and classical state spaces on which these structures act—specifically, via restriction to narrow wave packet states. Here, we describe a certain geometrical re-formulation and extension of the result that expectation values evolve approximately classically for narrow wave packet states, which relies essentially on the postulates of deformation quantization, but describes a relationship between the actions of quantum and classical algebras and groups over their respective state spaces that is non-trivially distinct from deformation quantization. The goals of the discussion are partly pedagogical in that it aims to provide a clear, explicit synthesis of known results; however, the particular synthesis offered aspires to some novelty in its emphasis on a certain general type of mathematical and physical relationship between the state spaces of different models that represent the same physical system, and in the explicitness with which it details the above-mentioned connection between quantum and classical models. (shrink)

A relation between physical consequences of the so-called Ehrenfest’s Paradox and the radial electric field E r (r) in the classical quasi-neutral tokamak plasma is shown. Basic author’s approach to the relativistic nature of the tokamak E r (r) has been described in Romannikov (J. Exp. Theor. Phys. 108:340–348, 2009). The experiment which can resolve the Ehrenfest’s Paradox is presented.

A generalization of Ehrenfest''s urn model is suggested. This will allow usto treat a wide class of stochastic processes describing the changes ofmicroscopic objects. These processes are homogeneous Markov chains. Thegeneralization proposed is presented as an abstract conditional (relative)probability theory. The probability axioms of such a theory and some simpleadditional conditions, yield both transition probabilities and equilibriumdistributions. The resulting theory interpreted in terms of particles andsingle-particle states, leads to the usual formulae of quantum and classicalstatistical mechanics; in terms of (...) chromosomes and allelic types, it allowsthe deduction of many genetical models including the Ewens sampling formula;in terms of agents'' strategies, it gives a justification of the ``herdbehaviour'''' typical of a population of heterogeneous economic agents. (shrink)

Our object in this paper is to study the antecedents, contents, implications, and impact of a not well-known or appreciated paper by EHRENFEST in 1911 on the essential nature of the different quantum hypotheses in radiation theory. After a careful analysis of EHRENFEST’s notebooks, correspondence, and publications, we conclude that the essential points of EHRENFEST’s paper were not perceived to a large extent, and hence that its implications were not considered thoroughly. Specifically, we show that EHRENFEST (...) contributed significantly to the clarifications of the differences between PLANCK’s and EINSTEIN’s respective quantum hypotheses, as well as to those of the concepts ‘‘discontinuity,’’ ‘‘quantization,’’ and ‘‘corpuscularity.’’. (shrink)

I discuss in detail the contents of the adiabatic hypothesis, formulated by Ehrenfest in 1916. I focus especially on the paper he published in 1916 and 1917 in three different journals. I briefly review its precedents and thoroughly analyze its reception until 1918, including Burgers’s developments and Bohr’s assimilation of them into his own theory. I show that until 1918 the adiabatic hypothesis did not play an important role in the development of quantum theory.

For a wave packet evolving quantum mechanically, the rates of change of the expectations and uncertainties of the position and momentum are exactly the same as if Wigner's function instantaneously obeyed a classical Liouville equation (whatever the Hamiltonian). This extension of Ehrenfest's theorem should be useful for dealing with the evolution and manipulation of quantum states.

We analyze the evolution of EHRENFEST's thought since he proved the necessity of quanta in 1911 until the formulation of his adiabatic hypothesis in 1914. We argue that his research contributed significantly to the solution of critical problems in quantum physics and led to a rigorous definition of the range of validity of BOLTZMANN's principle.

The first classification of general types of transition between phases of matter, introduced by Paul Ehrenfest in 1933, lies at a crossroads in the thermodynamical study of critical phenomena. It arose following the discovery in 1932 of a suprising new phase transition in liquid helium, the “lambda transition,” when W. H. Keesom and coworkers in Leiden, Holland observed a λhaped “jump” discontinuity in the curve giving the temperature dependence of the specific heat of helium at a critical value. This (...) apparent jump led Ehrenfest to introduce a classification of phase transitions on the basis of jumps in derivatives of the free energy function. This classification was immediately applied by A.J. Rutgers to the study of the transition from the normal to superconducting state in metals. Eduard Justi and Max von Laue soon questioned the possibility of its class of “second-order phase transitions” -- of which the “lambda transition was believed to be the arche type -- but C.J. Gorter and H.B.G. Casimir used an “order parameter to demonstrate their existence in superconductors. As a crossroads of study, the Ehrenfest classification was forced to undergo a slow, adaptive evolution during subsequent decades. During the 1940s the classification was increasingly used in discussions of liquid-gas, order-disorder, paramagnetic-ferromagnetic and normal-super-conducting phase transitions. Already in 1944 however, Lars Onsagers solution of the Ising model for two-dimensional magnets was seen to possess a derivative with a logarithmic divergence rather than a jump as the critical point was approached. In the 1950s, experiments further revealed the lambda transition in helium to exhibit similar behavior. Rather than being a prime example of an Ehrenfest phase transition, the lambda transition was seen to lie outside the Ehrenfest classification. The Ehrenfest scheme was then extended to include such singularities, most notably by A. Brain Pippard in 1957, with widespread acceptance. During the 1960s these logarithmic infinities were the focus of the investigation of “scaling” by Leo Kadanoff, B. Widom and others. By the 1970s, a radically simplified binary classification of phase transitions into “first-order” and “continuous” transitions was increasingly adopted. (shrink)

There seems to exist a dilemma in the literature as to the correct relativistic formula for the Sagnac phase-shift. The paper addresses this issue in the light of a novel, kinematically equivalent linear Sagnac-type thought experiment, which provides a vantage point from which the effect of rotation in the usual Sagnac effect can be analyzed. The question is shown to be related to the so-called rotating disc problem known as the Ehrenfest paradox. The relativistic formula for the Sagnac phase-shift (...) seems to depend on the way the paradox is resolved. Kinematic resolution of the Ehrenfest paradox proposed by some authors predicts the usually quoted formula for the Sagnac delay but the resolution itself is shown to be based upon some implicit assumptions regarding the behaviour of solid bodies under acceleration. In order to have a greater insight into the problem, a second version of the thought experiment involving linear motion of a “special type” of a non-rigid frame of reference is discussed. It is shown by analogy that the usually quoted special relativistic formula for the Sagnac delay follows, provided the material of the disc matches the “special type.”. (shrink)

This is an English translation of Ehrenfest's influential 1925 paper, 'Energy fluctuations in the radiation field or crystal lattice through superposition of quantized normal modes'.

En los orígenes de la física cuántica, a principios del siglo xx, la asimilación y maduración de las nuevas teorías requirió del establecimiento de vínculos con sus antecesoras, antes de poder desprenderse de ellas y formar una nueva estructura totalizadora de comprensión en esta disciplina. En este..

It is shown how a consistent kinematic resolution of Ehrenfest's paradox may be given in accordance with the special theory of relativity. Some statements by T. E. Phipps, Jr., connected with these matters, are commented upon. Problems connected with the relation between stress and strain are solved by a manifestly covariant formulation of Hooke's law.

Nous présentons une traduction française du compte rendu de L’Algèbre de la logique de Louis Couturat que Paul Ehrenfest rédigea en 1910 pour une revue scientifique russe. Dans ce texte, le physicien autrichien Ehrenfest propose d’appliquer la logique symbolique à l’analyse d’un réseau électrique. Dans une courte introduction, nous expliquons le contexte historique de ce texte et son importance philosophique.

In statistical thermodynamics the 2nd law is properly spelled out in terms of conditioned probabilities. As such it makes the statement, that `entropy increases with time' without preferring a time direction. In this paper we try to explain this statement---which is well known since the time of the Ehrenfests---in some detail within a systematic Bayesian approach.

Why is space 3-dimensional? The fi rst answer to this question, entirely based on Physics, was given by Ehrenfest, in 1917, who showed that the stability requirement for n-dimensional two-body planetary system very strongly constrains space dimensionality, favouring 3-d. This kind of approach will be generically called "stability postulate" throughout this paper and was shown by Tangherlini, in 1963, to be still valid in the framework of general relativity as well as for quantum mechanical hydrogen atom, giving the same (...) constraint for space{dimensionality. In the present work, before criticizing this methodology, a brief discussion has been introduced, aimed at stressing and clarifying some general physical aspects of the problem of how to determine the number of space dimensions. Then, the epistemological consequences of Ehrenfest's methodology are critically reviewed. An alternative procedure to get at the proper number of dimensions, in which the stability postulate - and the implicit singularities in three-dimensional physics - are not an essential part of the argument, is proposed. In this way, the main epistemological problems contained in Ehrenfest's original idea are avoided. The alternative methodology proposed in this paper is realized by obtaining and discussing the n-dimensional quantum theory as expressed in Planck's law, de Broglie relation and the Heisenberg uncertainty relation. As a consequence, it is possible to propose an experiment, based on thermal neutron di raction by crystals, to directly measure space dimensionality. Finally the distinguished role of Maxwell's electromagnetic theory in the determination of space dimensionality is stressed. (shrink)

Metascientific criteria used for explaining or constraining physical space dimensionality and their historical relationship to prevailing causal systems are discussed. The important contributions by Aristotle, Kant and Ehrenfest to the dimensionality of space problem are considered and shown to be grounded on different causal explanations: causa materialis for Aristotle, causa efficiens for young Kant and an ingenious combination of causa efficiens and causa formalis for Ehrenfest. The prominent and growing rôle played by causa formalis in modern physical approaches (...) to this problem is emphasized. (shrink)

We study the space geometry of a rotating disk both from a theoretical and operational approach; in particular we give a precise definition of the space of the disk, which is not clearly defined in the literature. To this end we define an extended 3-space, which we call “relative space:” it is recognized as the only space having an actual physical meaning from an operational point of view, and it is identified as the “physical space of the rotating platform.” Then, (...) the geometry of the space of the disk turns out to be non Euclidean, according to the early Einstein's intuition; in particular the Born metric is recovered, in a clear and self consistent context. Furthermore, the relativistic kinematics reveals to be self consistent, and able to solve the Ehrenfest's paradox without any need of dynamical considerations or ad hoc assumptions. (shrink)

Dirac’s identification of the quantum analog of the Poisson bracket with the commutator is reviewed, as is the threat of self-inconsistent overdetermination of the quantization of classical dynamical variables which drove him to restrict the assumption of correspondence between quantum and classical Poisson brackets to embrace only the Cartesian components of the phase space vector. Dirac’s canonical commutation rule fails to determine the order of noncommuting factors within quantized classical dynamical variables, but does imply the quantum/classical correspondence of Poisson brackets (...) between any linear function of phase space and the sum of an arbitrary function of only configuration space with one of only momentum space. Since every linear function of phase space is itself such a sum, it is worth checking whether the assumption of quantum/classical correspondence of Poisson brackets for all such sums is still self-consistent. Not only is that so, but this slightly stronger canonical commutation rule also unambiguously determines the order of noncommuting factors within quantized dynamical variables in accord with the 1925 Born-Jordan quantization surmise, thus replicating the results of the Hamiltonian path integral, a fact first realized by E.H. Kerner. Born-Jordan quantization validates the generalized Ehrenfest theorem, but has no inverse, which disallows the disturbing features of the poorly physically motivated invertible Weyl quantization, i.e., its unique deterministic classical “shadow world” which can manifest negative densities in phase space. (shrink)

The concept of rigid reference frame and of constricted spatial metric, given in the previous work [Class. Quantum Grav. 21, 3067 (2004)] are here applied to some specific space-times: in particular, the rigid rotating disc with constant angular velocity in Minkowski space-time is analyzed, a new approach to the Ehrenfest paradox is given as well as a new explanation of the Sagnac effect. Finally the anisotropy of the speed of light and its measurable consequences in a reference frame co-moving (...) with the Earth are discussed. (shrink)

The Ehrenfest paradox is reviewed in order to categorize classes of “resolution” that are logically at least semiconsistent. Cantoni's candidate is analyzed with reference to such a taxonomy of resolutions.

In 1909, Ehrenfest published a note in the Physikalische Zeitschrift showing that a Born rigid cylinder could not be set into rotation without stresses, as elements of the circumference would be contracted but not the radius. Ignatowski and Varićak challenged Ehrenfest’s result in the same journal, arguing that the stresses would emerge if length contraction were a real dynamical effect, as in Lorentz’s theory. However, no stresses are expected to arise, according to Einstein’s theory, where length contraction is (...) only an apparent effect due to an arbitrary choice of clock synchronization. Ehrenfest and Einstein considered this line of reasoning dangerously misleading and took a public stance in the Physikalische Zeitschrift, countering that relativistic length contraction is both apparent and real. It is apparent since it disappears for the comoving observer, but it is also real since it can be experimentally verified. By drawing on his lesser-known private correspondence with Varićak, this paper shows how Einstein used the Ehrenfest paradox as a tool for an ‘Einsteinian pedagogy’. Einstein’s argumentative stance is contrasted with Bell’s use of the Dewan-Beran thread-between-spaceships paradox to advocate for a ‘Lorentzian pedagogy’. The paper concludes that the disagreement between the two ways of ‘teaching special relativity’ stems from divergent interpretations of philosophical categories such as ‘reality’ and ‘appearance’. (shrink)

The thesis that observation necessarily fails to determine theory is false in the sense that observation can provide overwhelming evidence for a particular theory or even a hypothesis within the theory. The saga of quantum discontinuity illustrates the power of evidence to determine theory and shows how that power can be underestimated by inadequate caricatures of the evidential case. That quantum discontinuity can save the phenomena of black body radiation is the widely known result, but it leaves open the possibilities (...) of other accounts. That these phenomena, with the aid of minimal assumptions, entail quantum discontinuity is the crucial but now largely forgotten result. It was first demonstrated by Ehrenfest and Poincaré in 1911 and 1912. (shrink)

We revisit the analogy suggested by Madelung between a non-relativistic time-dependent quantum particle, to a fluid system which is pseudo-barotropic, irrotational and inviscid. We first discuss the hydrodynamical properties of the Madelung description in general, and extract a pressure like term from the Bohm potential. We show that the existence of a pressure gradient force in the fluid description, does not violate Ehrenfest’s theorem since its expectation value is zero. We also point out that incompressibility of the fluid implies (...) conservation of density along a fluid parcel trajectory and in 1D this immediately results in the non-spreading property of wave packets, as the sum of Bohm potential and an exterior potential must be either constant or linear in space. Next we relate to the hydrodynamic description a thermodynamic counterpart, taking the classical behavior of an adiabatic barotopric flow as a reference. We show that while the Bohm potential is not a positive definite quantity, as is expected from internal energy, its expectation value is proportional to the Fisher information whose integrand is positive definite. Moreover, this integrand is exactly equal to half of the square of the imaginary part of the momentum, as the integrand of the kinetic energy is equal to half of the square of the real part of the momentum. This suggests a relation between the Fisher information and the thermodynamic like internal energy of the Madelung fluid. Furthermore, it provides a physical linkage between the inverse of the Fisher information and the measure of disorder in quantum systems—in spontaneous adiabatic gas expansion the amount of disorder increases while the internal energy decreases. (shrink)

I show explicitly how concerns about wave function collapse and ontology can be decoupled from the bulk of technical analysis necessary to recover localized, approximately Newtonian trajectories from quantum theory. In doing so, I demonstrate that the account of classical behavior provided by decoherence theory can be straightforwardly tailored to give accounts of classical behavior on multiple interpretations of quantum theory, including the Everett, de Broglie-Bohm and GRW interpretations. I further show that this interpretation-neutral, decoherence-based account conforms to a general (...) view of inter-theoretic reduction in physics that I have elaborated elsewhere, which differs from the oversimplified and often ambiguous picture that treats reduction simply as a matter of taking limits. This interpretation-neutral account rests on a general three-pronged strategy for reduction between quantum and classical theories that combines decoherence, an appropriate form of Ehrenfest's Theorem, and a decoherence-compatible mechanism for collapse. It also incorporates a novel argument as to why branch-relative trajectories should be approximately Newtonian, which is based on a little-discussed extension of Ehrenfest's Theorem to open systems, rather than on the more commonly cited but less germane closed-systems version. In the Conclusion, I briefly suggest how the strategy for quantum-classical reduction described here might be extended to reduction between other classical and quantum theories, including classical and quantum field theory and classical and quantum gravity. (shrink)

The specific heat of hydrogen gas at low temperatures was first measured in 1912 by Arnold Eucken in Walther Nernst’s laboratory in Berlin, and provided one of the earliest experimental supports for the new quantum theory. Even earlier, Nernst had developed a quantum theory of rotating diatomic gas molecules that figured in the discussions at the first Solvay conference in late 1911. Between 1913 and 1925, Albert Einstein, Paul Ehrenfest, Max Planck, Fritz Reiche, and Erwin Schrödinger, among many others, (...) attempted theoretical descriptions of the rotational specific heat of hydrogen, with only limited success. Quantum theory also was central to the study of molecular spectra, where initially it was more successful. Moreover, the two problems interacted in sometimes surprising ways. Not until 1927, following Werner Heisenberg’s discovery of the behavior of indistinguishable particles in modern quantum mechanics, did American theorist David Dennison find a successful theory of the specific heat of hydrogen. (shrink)

In a rarely quoted paper, published in 1958 in the American Journal of Physics, T. Ehrenfest-Afanassjewa introduced the idea that the concept of chance as employed in physics is subject to what she called a ‘Limited Belief in Chance’. In this paper I elaborate the latter concept and the distinction between absolute chance and relative randomness, where the latter, but not the former, is governed by the theory of probability. I argue that in the twentieth century virtually nobody believes (...) seriously in the possiblity of absolute chance, whereas the concept of chance in the Scientific World Picture is only ‘chancy’ relative to a limited belief in chance grounded in the Manifest World Picture of an orderly world. (shrink)

This paper discusses an argument by Norton to the effect that reversible processes in thermodynamics have paradoxical character, due to the infinite-time limit. For Norton, one can “dispel the fog of paradox” by adopting a distinction between idealizations and approximations, which he himself puts forward. Accordingly, reversible processes ought to be regarded as approximations, rather than idealizations. Here, we critically assess his proposal. In doing so, we offer a resolution of his alleged paradox based on the original work by Tatiana (...) Ehrenfest-Afanassjeva on the foundations of thermodynamics. (shrink)

In his 1916 review paper on general relativity, Einstein made the often-quoted oracular remark that all physical measurements amount to a determination of coincidences, like the coincidence of a pointer with a mark on a scale. This argument, which was meant to express the requirement of general covariance, immediately gained great resonance. Philosophers such as Schlick found that it expressed the novelty of general relativity, but the mathematician Kretschmann deemed it as trivial and valid in all spacetime theories. With the (...) relevant exception of the physicists of Leiden (Ehrenfest, Lorentz, de Sitter, and Nordström), who were in epistolary contact with Einstein, the motivations behind the point-coincidence remark were not fully understood. Only at the turn of the 1960s did Bergmann (Einstein’s former assistant in Princeton) start to use the term ‘coincidence’ in a way that was much closer to Einstein’s intentions. In the 1980s, Stachel, projecting Bergmann’s analysis onto his historical work on Einstein’s correspondence, was able to show that what he started to call ‘the point-coincidence argument’ was nothing but Einstein’s answer to the infamous ‘hole argument’. The latter has enjoyed enormous popularity in the following decades, reshaping the philosophical debate on spacetime theories. The point-coincidence argument did not receive comparable attention. By reconstructing the history of the argument and its reception, this paper argues that this disparity of treatment is not justified. This paper will also show that the notion that only coincidences are observable in physics marks every critical step of Einstein’s struggle with the meaning of coordinates in physics. (shrink)

A comparison is made of the traditional Loschmidt and Zermelo objections to Boltzmann's H-theorem, and its simplified variant in the Ehrenfests' 1912 wind-tree model. The little-cited 1896 objection of Zermelo is also analysed. Significant differences between the objections are highlighted, and several old and modern misconceptions concerning both them and the H-theorem are clarified. We give particular emphasis to the radical nature of Poincare's and Zermelo's attack, and the importance of the shift in Boltzmann's thinking in response to the objections (...) as a whole. (shrink)

Textbooks in quantum mechanics frequently claim that quantum mechanics explains the success of classical mechanics because “the mean values [of quantum mechanical observables] follow the classical equations of motion to a good approximation,” while “the dimensions of the wave packet be small with respect to the characteristic dimensions of the problem.” The equations in question are Ehrenfest’s famous equations. We examine this case for the one-dimensional motion of a particle in a box, and extend the idea deriving a special (...) case of the ideal gas law in terms of the mean value of a generalized force, which has been used in statistical mechanics to define ‘pressure’. The example may be an important test case for recent philosophical theories about the relationship between micro-theories and macro-theories in science. (shrink)

Due to its extraordinary predictive power and the great generality of its mathematical structure, quantum theory is able, at least in principle, to describe all the microscopic and macroscopic properties of the physical world, from the subatomic to the cosmological level. Nevertheless, ever since the Copen hagen and Gottingen schools in 1927 gave it the definitive formu lation, now commonly known as the orthodox interpretation, the theory has suffered from very serious logical and epistemologi cal problems. These shortcomings were immediately (...) pointed out by some of the principal founders themselves of quantum theory, to wit, Planck, Einstein, Ehrenfest, Schrodinger, and de Broglie, and by the philosopher Karl Popper, who assumed a position of radical criticism with regard to the standard formulation of the theory. The aim of the participants in the workshop on Open Questions in Quantum Physics, which was held in Bari (Italy), in the Department of Physics of the University, during May 1983 and whose Proceedings are collected in the present volume, accord ingly was to discuss the formal, the physical and the epistemo logical difficulties of quantum theory in the light of recent crucial developments and to propose some possible resolutions of three basic conceptual dilemmas, which are posed respectively ~: (a) the physical developments of the Einstein-Podolsky-Rosen argument and Bell's theorem, i. e. (shrink)

En mécanique statistique, un système physique est représenté par un système mécanique avec un très grand nombre de degrés de liberté. Ce qui est expérimentalement accessible, croit-on, se limite à des moyennes temporelles sur de longues périodes. Or, il est bien connu qu’un système physique tend vers un équilibre thermodynamique. Ainsi, les moyennes temporelles censées représenter les résultats de mesure doivent être indépendantes du temps. C’est pourquoi elles sont associées à des temps infinis. Ces moyennes sont par contre difficilement analysables, (...) et c’est pourquoi la moyenne des phases est utilisée. La justification de l’égalité de la moyenne temporelle infinie et de la moyenne des phases est le problème ergodique. Ce problème, sous une forme ou une autre, a fait l’objet d’études de la part de Boltzmann, les Ehrenfest, Birkhoff, Khinchin, et bien d’autres, jusqu’à devenir une théorie à part entière en mathématique. Mais l’introduction de temps infinis pose des problèmes physiques et philosophiques d’importance. En effet, si l’infini a su trouver une nouvelle place dans les mathématiques cantoriennes, sa place en physique n’est pas aussi assurée. Je propose donc de présenter les développements conceptuels entourant la théorie ergodique en mécanique statistique avant de me concentrer sur les problèmes épistémologiques que soulève la notion d’infini dans ces mêmes développements. (shrink)

Newtonian and Scr{\"o}dinger dynamics can be formulated in a physically meaningful way within the same Hilbert space framework. This fact was recently used to discover an unexpected relation between classical and quantum motions that goes beyond the results provided by the Ehrenfest theorem. A formula relating the normal probability distribution and the Born rule was also found. Here the dynamical mechanism responsible for the latter formula is proposed and applied to measurements of macroscopic and microscopic systems. A relationship between (...) the classical Brownian motion and the diffusion of state on the space of states is discovered. The role of measuring devices in quantum theory is investigated in the new framework. It is shown that the so-called collapse of the wave function is not measurement specific and does not require a ``concentration" near the eigenstates of the measured observable. Instead, it is explained by the common diffusion of state over the space of states under interaction with the apparatus and the environment. This in turn provides us with a basic reason for the definite position of macroscopic bodies in space. (shrink)

Quantum observables can be identified with vector fields on the sphere of normalized states. The resulting {\it vector representation} is used in the paper to undertake a simultaneous treatment of macroscopic and microscopic bodies in quantum mechanics. Components of the velocity and acceleration of state under Schr{\"o}dinger evolution are given a clear physical interpretation. Solutions to Schr{\"o}dinger and Newton equations are shown to be related beyond the Ehrenfest results on the motion of averages. A formula relating the normal probability (...) distribution and the Born rule is found. (shrink)

The problem of relation between statistical mechanics (SM) and classical mechanics (CM), especially the question whether SM can be founded on CM, has been a subject of controversies since the rise of classical statistical mechanics (CSM) at the end of 19th century. The first views rejecting explicitly the possibility of laying the foundations of CSM in CM were triggered by the "Wiederkehr-" and "Umkehreinwand" arguments. These arguments played an important role in the debate about Boltzmann's original H-theorem and led to (...) the so called statistical H-theorem proposed by Boltzmann himself. (For the history of these early debates we refer to Brush's monograph (Brush 1976).) After CSM had been brought to "canonical form" by the Ehrenfests, (Ehrenfest and Ehrenfest 1959) the physicists turned away from the foundational problem leaving it to mathematicians to worry about in the form of what has become called the ergodic theory. In retrospect, the physicists' general mood seems to have been the hope that ergodic theory establishes rigorously what is needed to found CSM on CM and which had been expressed essentially by Boltzmann already (Wightman 1985). However, very few physicists followed closely the developments in the mathematical theory of dynamic systems. One of those who did was the Russian physicist N.S. Krylov. (For a brief description of Krylov's personal life we refer to the papers in (Krylov 1979).). (shrink)

This paper presents a new foundation of equilibrium thermodynamics based on certain ideas of T. Ehrenfest. The main result is the proof for the existence of entropy as a consequence of the conservation of energy for conservative thermal systems.

In this article, we analyze the third of three papers, in which Einstein presented his quantum theory of the ideal gas of 1924–1925. Although it failed to attract the attention of Einstein’s contemporaries and although also today very few commentators refer to it, we argue for its significance in the context of Einstein’s quantum researches. It contains an attempt to extend and exhaust the characterization of the monatomic ideal gas without appealing to combinatorics. Its ambiguities illustrate Einstein’s confusion with his (...) initial success in extending Bose’s results and in realizing the consequences of what later came to be called Bose–Einstein statistics. We discuss Einstein’s motivation for writing a non-combinatorial paper, partly in response to criticism by his friend Ehrenfest, and we paraphrase its content. Its arguments are based on Einstein’s belief in the complete analogy between the thermodynamics of light quanta and of material particles and invoke considerations of adiabatic transformations as well as of dimensional analysis. These techniques were well known to Einstein from earlier work on Wien’s displacement law, Planck’s radiation theory and the specific heat of solids. We also investigate the possible role of Ehrenfest in the gestation of the theory. (shrink)

In recent previous work, the authors proposed a new approach extending the framework of statistical mechanics to reparametrization-invariant systems with no additional gauges. In this paper, the approach is generalized to systems defined by more than one Hamiltonian constraint. We show how well-known features as the Ehrenfest–Tolman effect and the Jüttner distribution for the relativistic gas can be consistently recovered from a covariant approach in the multi-fingered framework. Eventually, the crucial role played by the interaction in the definition of (...) a global notion of equilibrium is discussed. (shrink)

The aim of the present paper is to give a purely probabilistic account for the approach to equilibrium of classical and quantum gas. The probability function used is classical. The probabilistic dynamics describes the evolution of the state of the gas due to unary and binary collisions. A state change amounts to a destruction in a state and the creation in another state. Transitions probabilities are splittled into destructions terms, denoting the random choice of the colliding particle(s), and creation terms, (...) describing the allocation of the same particle(s) on the final state(s). While the destruction term is the same for all types of particles, the creation one depends upon a parameter bound to the interpraticle correlation. The transition probabilities give rise to a homogeneous Markov chain. The equilibrium distributions satisfy the principle of detailed balance. Relaxation times depend upon the interparticle correlation. Relationships with the Ehrenfest urn model, Brillouin unified method, ensemble interpretation, and quantum H-theorem are considered too. (shrink)

The dynamical equations for black-body radiation are derived from the vanishing of the stress-energy tensor and shown to imply a theorem which generalizes that of Tolman and Ehrenfest[Phys. Rev. 36,1791 (1930)]. A method for finding the general solution of these equations is given.

Ludwig Boltzmann’s work in theoretical physics exhibits an approach to the construction of theory that he transmitted to the succeeding generation by example. It involved the construction of clear models, allowed more than one, and was not based solely on the existing facts, with the intent of examining and criticizing the assumptions that made each model work. This tacit program influenced physicists like Ehrenfest and Einstein and the philosopher Wittgenstein, suggesting ways that they used to make further advances.

L'idea di relazionare la dimensionalità dello spazio ad una legge fisica, contenuta nel primo scritto di Kant "Pensieri sulla veridica estima delle forze vive", svela un modo di guardare i rapporti tra Fisica e Matematica così nuovo ed originale che potè essere sviluppato e compreso nella sua plenitudine soltanto nel secolo XX. Ci riferiamo qui ala prospettiva aperta da Ehrenfest nel suo "In what way does it become manifest in the fundamental laws of physics that space has three dimensions?". (...) In questo saggio si cerca di comprendere alcune delle radici fisiche, filosofiche e teologiche dell’approccio di Kant a questo problema, e la sua inserzione nel dibattito delle idee che sostengono la concezione meccanicista del Mondo. In primo luogo, si sottolinea che da una lettura attenta del suo Gedanken von der wahren Schätzung der lebendigen Kräfte si può concludere che - al contrario di quanto viene normalmente creduto - Kant non riuscì a dimostrare la sua congettura che la tridimensionalità dello spazio possa essere fondata sulla legge di gravitazione di Newton, secondo la quale la forza tra due corpi va come l'inverso del quadrato della distanza che li separa; in realtà, egli si limitò a giustificare la tridimensionalità della estensione. Dopo aver considerato la distinzione fra estensione e spazio, si fa notare che la catena logica della congettura del giovane Kant, presuppone la forza come concetto basilare, intrinseco alle sostanze. È mediante questa forza che si stabiliscono le relazioni tra le sostanze, le quali, a sua volta, creano l'ordine su cui viene basato il concetto di spazio. Si deve badare che la coerenza del suo argomento richiede, in ultima analisi, concepire la percezione, cioè, l'interazione fra le sostanze e l'anima, come risultato della medesima forza di lunga portata. Inoltre, si dimostra che l'argomento di Kant non poteva essere costrutto su una intelaiatura puramente newtoniana nel 1747. Infatti, egli costruì sua dimostrazione a partire da una concezione del mondo basata sul legato di Cartesio (soprattutto l'idea di um ordine cosmico) e Leibniz (l'esistenza di una forza che precede l'estensione e l'idea di spazio relativo) alla quale sovrappone, in modo ibrido, la legge di gravitazione di Newton. Si argomenta che questa scelta fu fortemente influenzata da alcune idee teologiche del Giovane Kant che divergono da quelle di Cartesio, Newton e Leibniz. Fu questo carattere ibrido del suo approccio che permise a Kant di trattare la questione della dimensionalità dello spazio come un problema di fisica, ma, che, allo stesso tempo, a causa delle sue limitazioni intrinsiche, portò Kant a provare appena la tridimensionalità della estensione. (shrink)

Space is one of the most fundamental concepts over which scientific knowledge has been constructed. But it is also true that space concepts extrapolate by far the scientific domain, and permeate many other branches of human knowledge. Those are fascinating aspects that could di per se justify the compilation of a long bibliography. Another one is the passion for books. My interest in some physical, historical and philosophical problems concerning the concept of space in Physics, and its properties, can be (...) traced back to the early 1980. Since that time, I have being studying, with several collaborators, the influence of space dimensionality in different physical phenomena, like the Casimir Effect, neutron diffraction, Cosmic Microwave Background Radiation and the stability of Schrödinger’s and Dirac’s hydrogen atom in arbitrary number of dimensions, as well as epistemological aspects of the works of Kant, Ehrenfest an others on this particular subject. Meanwhile, I gave lectures about the History of the Concepts of Space in Physics at the Centro Brasileiro de Pesquisas Físicas (CBPF), at the Program of History of Sciences, Technics and Epistemology of the Federal University of Rio the Janeiro (UFRJ) and also at the Physics Institute of the State University of Rio de Janeiro (UERJ). As a consequence of both this interest and my love for books, I continuously bought books on space for my personal library which contains now quite one half of all the books quoted here. In 1996, Roberto Moreira and I published the Sources for the History of Space concepts in Physics: From 1845 to 1995 in the series Notas de Física of the CBPF (NF # 084/96), which is still available online. This was a first initiative to share our bibliography on space. At that time, it contained 1075 references, including 414 books entirely devoted to space, 380 articles in periodical journals and proceedings and 281 miscellaneous citations. After fifteen years, I decided to focus my attention here to collect what in my opinion are the 601 essential books which could be useful for anyone interested on learning about space. An important restriction is imposed here by language: only texts written in English, French, German, Italian, Spanish, Portuguese and Latin were considered. References are given in chronological order covering the period between 1739 and 2010. For each year, the books are listed in alphabetical order of authors’ name. An onomastic index is included at the end of the book. -/- Francisco Caruso Rio de Janeiro, October 2011 . (shrink)