Abstract
This Lecture has two purposes. The first is to demonstrate that one can obtain useful results by making formal manipulations of the partition function and the ensemble average. The results of these manipulations include the First and Second Generalized Equipartition Theorems and the relation between energy fluctuations and the specific heat. The second purpose of the lecture is to illustrate the utility of studying special cases, as a method for checking one’s theoretical understanding. To show why special cases are helpful, we derive a result that is less general than it at first appears. The apparent result will be tested unsuccessfully against a special case; insight gained from failure will be used to derive a fully general result.
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References
For example, a fully microscopic analysis of the dielectric susceptibility of a solid only obtains a local, intensive form for if the shape of V is included in the calculation, the shape finally being divided out of the answer.
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For more details on this Hamiltonian, which describes ions in some ferrolectric crystals, see A. Morita and D. G. Frood, J. Phys. D 11, 2409 (1978) and A. D. Booth, J. Comp. Phys. 46, 423 (1982). Recall that rapidity of convergence of power series is in practical terms much less important now than was the case even five or ten years ago.
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© 2000 Springer-Verlag Berlin Heidelberg
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Phillies, G.D. (2000). Formal Manipulations of the Partition Function. In: Elementary Lectures in Statistical Mechanics. Graduate Texts in Contemporary Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1264-5_9
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DOI: https://doi.org/10.1007/978-1-4612-1264-5_9
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-7068-3
Online ISBN: 978-1-4612-1264-5
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