Abstract
In this Chapter we discuss the issue of equivalence between the microcanonical and the canonical ensemble. Depending on the system under study the two ensembles may display various degrees of equivalence ranging from full equivalence to inequivalence. We illustrate that with examples.
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Notes
- 1.
This is in fact true for \(\Psi _c\) as well, see Eq. (4.35).
- 2.
It must be remarked that from a practical point of view, the evaluation of \(\Omega \) is more straightforward than that of \(\omega \).
- 3.
Note that these quantities do not depend on \(\epsilon \).
- 4.
The symbol \(s_{ij..}\) stands for the partial derivative of s with respect to its i-th argument, followed by the partial derivative with respect to its j-th argument, and so on.
References
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Boltzmann, L.: Sitzungberichte der Kaiserlichen Akademie der Wissenschaften. Mathematisch-Naturwissen Classe. Abt. II, LXXVI (1877), pp 373–435 (Wien. Ber. 1877, 76:373–435). Reprinted in Wiss. Abhandlungen, Vol. II, reprint 42, pp 164–223, Barth, Leipzig, 1909. Translated in English by K. Sharp and F. Matschinsky, Entropy 17(4), 1971–2009 (2015). https://doi.org/10.3390/e17041971
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Campisi, M. (2021). Ensemble (in)-Equivalence. In: Lectures on the Mechanical Foundations of Thermodynamics. SpringerBriefs in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-87163-5_7
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DOI: https://doi.org/10.1007/978-3-030-87163-5_7
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