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1

Why must we work in the phase space?

Sławianowski J. J., Schroeck Jr F. E., Martens A.

IPPT Reports on Fundamental Technological Research

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2016

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nr 1

1--162

EN

We are going to prove that the phase-space description is fundamental both in the classical and quantum physics. It is shown that many problems in statistical mechanics, quantum mechanics, quasi-classical theory and in the theory of integrable systems may be well-formulated only in the phase-space language. There are some misunderstandings and confusions concerning the concept of induced probability and entropy on the submanifolds of the phase space. First of all, they are restricted only to hypersurfaces in the phase space, i.e., to the manifolds of the defect of dimension equal to one. But what is more important, it was assumed there that the phase-space geometry was metrical-Euclidean and the resulting metric geometry of the microcanonical ensemble was obtained by the reduction of the primary Euclidean geometry to the corresponding submanifold. But it is well-known that the phase-space manifold has no natural metric geometry and that all concepts to be used must be of symplectic origin. Otherwise they are just accidental or artificial. So, instead we show that even if the configuration space is endowed with some metric, then in general the true geometry of submanifolds in the corresponding cotangent bundle (phase-space) is of different origin which has nothing to do with the mentioned configuration space Riemannian geometry, instead it is of purely symplectic origin. And this is sufficient to constructing microcanonical ensemble and entropy concepts. In any case, the purely symplectic phase-space geometry is sufficient to obtain everything within the completely metric-free language.

PL

Chcemy wykazać, że opis zjawisk mechanicznych oparty na pojęciu przestrzeni fazowej jest fundamentalny zarówno z klasycznego jak i kwantowego punktu widzenia. Pokazujemy, że liczne problemy mechaniki statystycznej, teorii kwantów i mechaniki quasiklasycznej oraz teorii układów całkowalnych mogą być dobrze sformułowane wyłącznie w języku symplektycznej przestrzeni fazowej. Istnieje mnóstwo nieporozumień czy wręcz błędów dotyczących pojęcia prawdopodobieństwa warunkowego i entropii w przypadku podrozmaitości przestrzeni fazowej. Przede wszystkim są one zazwyczaj definiowane dla przypadku powierzchni o defekcie wymiaru jeden. Co jednak dużo ważniejsze, zwykle zakłada się, że przestrzeń fazowa ma jednocześnie metryczną geometrię Euklidesową. Geometria metryczna podrozmaitości, używana w konstrukcji zespołu mikrokanonicznego, jest otrzymywana jako redukcja, ograniczenie pierwotnej geometrii Euklidesowej. Wiadomo jednak, że rozmaitość przestrzeni fazowej nie ma żadnej „wrodzonej” geometrii metrycznej i że wszystkie podstawowe pojęcia, wyjąwszy dynamikę konkretnych modeli, muszą mieć czysto symplektyczną genezę. W przeciwnym wypadku są one przypadkowe lub wręcz sztuczne. Zatem, nawet jeśli wyjściowa przestrzeń konfiguracyjna ma zadaną geometrię typu metrycznego, to na ogół właściwa geometria podrozmaitości w wiązce ko-stycznej, przynajmniej ta istotna dla pojęć statystycznych, nie jest związana z metryką konfiguracyjną i ma czysto symplektyczną genezę. I to wystarcza dla skonstruowania pojęcia zespołu mikrokanonicznego i entropii. W każdym razie, czysto symplektyczna geometria przestrzeni fazowej wystarcza do otrzymania pojęć mechaniki statystycznej w obrębie języka całkowicie niemetrycznego. W przypadku, gdy przestrzeń konfiguracyjna jest Euklidesowa, implikowane przez metrykę pojęcia statystyczne pokrywają się z symplektycznymi. W ogólnym wypadku nie musi tak być. Pokazujemy, że pojęcia te dadzą się wprowadzić w języku czysto symplektycznym, niezależnym od metryki konfiguracyjnej. Dotyczy to także uogólnionych rozkładów mikrokanonicznych.

2

Dynamical systems with internal degrees of freedom in non-Euclidean speces

Sławianowski J. J., Kovalchuk V., Gołubowska B., Martens A., Rożko E. E.

Prace Instytutu Podstawowych Problemów Techniki PAN

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2006

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nr 8

1-121

EN

The primary concept of Newton mechanics is that of the material point moving in three-dimensional Euclidean space. A good deal of the theory depends only on the affine sector of geometry. The metric structure becomes essential when constructing particular functional models of forces; the concepts of energy, work, and power (time rate of work) also depend in an essential way on the metric tensor. The Galilei relativity principle implies that,l as a matter of vactl, it is not three-dimencional Euclidean space but rather four-dimensional Galilean space-time that is a proper arena of mechanics. This space-time has relatively complicated structure, does not carry any natural four-dimensional metric tensor and fails to be the Cartesian product of space and time. There exists the absolute time, but the absolute space does not. In the sequel we concentrate onf the other kind of problems, so the analysis of the subtle space-time aspects will be almost absent in our treatment. Newton theory becomes essentially realistic and viable when multiparticle systems are analyzed. It is just there where metrical concepts become almost unavoidable, because it is practically impossible to construct any realistic model of interparticle forces without the explicit use of the metric tensor. Extended bodies are described as discrete or continuous systems of material points. Their motion consists of that of the center of mass, i.e., translational motion and the relative motion of constituents with respect to the center of mass. The total configuration space may be identified with the Cartesian product of the physical space (translational motion) and the configuration space of relative motion. In many physical problems the structure of mutual interactions leads to certain hierarchy of degrees of freedom of the relative motion; in particular, some constraints may appear. The effective configuration space becomes then the Cartesian product ot the physical space and some manifold of additional degrees of freedom. There are situations when this auxiliary manifold and the corresponding dynamics are postlulated as something rather primary then derived from the multiparticle models. Usually the guiding hints are based on some symmetry principles. In this way the concept of internal degrees of freedom replaces that of relative motion. Sometimes it is a merely convenient procedure, but one can also admit something like essentially internal degrees of freedom not derivable from any multiparticle mode. After all, the very concept of the material point is an abstraction of a small piece of matter.

3

Invariant geodetic systems on Lie groups and affine models of internal and collective degrees of freedom

Sławianowski J. J., Kovalchuk V., Sławianowska A., Gołubowska B., Martens A., Rożko E. E., Zawistowski Z. J.

Prace Instytutu Podstawowych Problemów Techniki PAN

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2004

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nr 7

3-164

EN

In some of our earlier papers including rather old ones we have discussed the concept of affinely-rigid body, i.e., continuous, discrete, or simply finite system of material points subject to such constraints that all affine relations between its elements are frozen during any admissible motion. For example, all material straight lines remain straight lines in the course of evolution, and their parallelism is also a constant, non-violated property. Unlike this, the metrical features, like distances and angles, need not be preserved. In other words, such a body is restricted in its behaviour to rigid translations, rigid rotations, and homogeneous deformations. Models of this kind may be successfully applied in a very wide spectrum of physical problems like nuclear dynamics (droplet model of the atomic nuclei), molecular vibrations, macroscopic elasticity (in situations when the length of excited waves is comparable with the size of the body), in the theory of microstructured bodies (micromorphic continua), in geophisics (the theory of the shape of Earth), and even in large-scale astropysics (vibrating stars, vibrating concentrations of the cosmic substratum, like galaxies or concentrations of the interstellar dust).

4

Selective hydrogenation of acetylene in an ethylene stream in an adiabatic packed bed reactor

Westerterp K.R., Bos A.N.R., Wijngaarden R.J., Kusters W.C., Martens A.

Inżynieria Chemiczna i Procesowa

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2000

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T. 21, z. 1

7-28

EN

In earlier studies, the behavior of single catalyst pellets of palladium on alumina was investigated for the reaction of acetylene in an ethylene stream with hydrogen. Particle runaway, temperature over- and undershoots and chemically induced temperature oscillations have been observed. After that, the steady state and dynamic behavior of an adiabatic packed bed reactor has been studied experimentally. Temperature profiles of both the gas and solid phase as well as local temperature differences between the two phases were measured. Also here the temperature in the reaction zone exhibited oscillatory behavior. On addition of CO, the oscillations disappeared and the selectivity improved. For a given set of operating conditions there existed a relatively small range of CO contents with good selectivity and satisfactory conversion. This range depends strongly on the inlet temperature. The dynamic response of the reactor to changes in the CO content showed a considerable wrong-way behavior. This high sensitivity to fluctuations in the CO content, found for our experimental reactor, indicates a probable cause for a thermal runaway in industrial practice. Recommendations for a stable reactor operation are given.