Download Encyclopedia of Physics. Principles of Thermodynamics and by S. Flügge PDF

By S. Flügge

177 Die statistische Mechanik ist ursprunglich von BOLTZMANN und GIBBS auf der Grundlage der Hamiltonschen Mechanik entwickelt worden. Da wir jedoch den Ausgangspunkt als ein challenge der Atom-Mechanik formullert haben, muB eine streng logische Darstellung des Gebietes notwendig an die Quantenmechanik anknupfen. Trotzdem werden wir im folgenden aus zwei Griinden die Theorie zuerst auf klassischer Grundlage entwickeln. Einmal ist das Begriffssystem der Quantenstatistik (abnlich wie das der Quantenmechanik) in moglichst weit gehender Analogie zur klassischen Theorie aufgebaut worden. Die Vorwegnahme der letzteren bringt daher eine wesentliche Vereinfachung der Darstellung und eine Erleichterung des Verstandnisses. Zum anderen bleibt fUr die uberwiegende Mehrzahl der Anwendungen von der Quantenstatistik nur eine geringfUgige Korrektur an den Resultaten der klassischen Theorie ubrig, die sich bereits im Rahmen der letzteren plausibel machen (wenn auch nicht beweisen) laBt. Prak tisch benutzt guy daher in den meisten Fillen die Methoden der klassischen statistischen Mechanik, deren Kenntnis somit ohnehin unentbehrlich ist. Die axiomatische foundation der statistischen Mechanik wird naturgemaB zunachst durch die Axiome der Wahrscheinlichkeitsrechnung und der Mechanik gebildet. Auf dieser Grundlage lassen sich allgemeine Satze uber statistische Gesamtheiten, ableiten, die fur den Aufbau der Theorie von auBerordentlicher Bedeutung sind, aber im wesentlichen formalen Charakter besitzen. Bei dem Versuch, die statisti sche Mechanik zu einer physikalischen Theorie auszugestalten, stoBt guy auf eine Lucke, die sich trotz vielen Bemuhungen bisher auf deduktivem Wege nicht hat schlie Ben lassen.

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Extra resources for Encyclopedia of Physics. Principles of Thermodynamics and Statistics

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8) to structural parameterization in vector autoregressive models was considered in Ahn (1988). 1 Causality The problem of causal relationship has been a fascinating subject for both philosophers and statisticians for centuries. In statistics, when a student first comes across simple correlation analysis, he is usually cautioned that a significant cross-correlation does not necessarily imply a cause and effect type relationship. On the other hand, it is difficult to define clearly what causality means.

We have the innovation outliers (IO) if the noise process at has a heavy tailed distribution compared with the normal distribution. This type of outlier is less problematic if at has finite fourth order moment. 5). Another more serious type of outliers is known as the additive outliers (AO). 1), and Wt is a contaminating process with P (Wt = 0) = C for some C with 0 ≤ C ≤ 1. The presence of Wt masks the original autocorrelation structure of Xt and hence causes greater problems in the modeling of Xt .

1 that the observations at t = 83 and 84 are somewhat larger than the rest of the series and may be regarded as outliers. 2 the series can be identified as an AR(1) process because the partial autocorrelation has a cut-off after lag 1. The two observations were then replaced by observations from the same months in the previous year (1971). 4 gives © 2004 by Chapman & Hall/CRC the sample autocorrelations and the partial autocorrelation. It can be seen that the dependence structure of the series is now much stronger and the partial autcorrelations suggested an AR(2) model instead.

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