Abstract
The cluster ensemble is inherently discrete but when the characteristic cluster size is much larger than the unit of the ensemble, the MPD may be treated as a continuous variable. We define the continuous limit by the condition
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Notes
- 1.
We use the notation J[f] to indicate J is a functional of function f. The calculus of functionals is reviewed briefly in the appendix of Chap. 7.
- 2.
- 3.
The properties of the intensive partition function for the generalized ensemble (M 1, M 2, ⋯ ; N) were discussed in Sect. 3.8.2. Here we adapt the results of that section to the simple microcanonical ensemble \(\mu \mathcal C(M,N)\) with \(M={\bar x} N\).
- 4.
The term that is dropped is \(\log \sqrt {2 \pi N {\bar x} ({\bar x}-1)}\). This amounts to using the simpler Stirling formula, \(\log x!=x\log x - x\).
- 5.
When \({\bar x}\gg 1\) the right-hand side of Eq. (4.17) is asymptotically equal to the derivative of \({\bar x}\log {\bar x}\):
$$\displaystyle \begin{aligned} \log\omega\to \frac{d {\bar x}\log{\bar x}}{d{\bar x}} = \log{\bar x}+1 . \end{aligned}$$Then obtain β as \(d\log \omega /d{\bar x}\) and \(\log q\) as \(\log \omega -\beta {\bar x}\).
- 6.
Start with Eq. (4.14)
$$\displaystyle \begin{aligned} q = \mathcal L\big\{w;x;\beta\big\} , \end{aligned}$$and solve for w by inverting the transform:
$$\displaystyle \begin{aligned} w = \mathcal L^{-1}\big\{q;\beta;x\big\} . \end{aligned}$$ - 7.
- 8.
The notation on the right-hand side means that y is treated as a constant during the Laplace transform and is to be replaced by \({\bar x}\) only after the transform is completed. If y is replaced by \({\bar x}\) inside the transform and we use \({\bar x}=2/\beta \), the result of the inverse transform is not \(w(x;{\bar x})\) but w ∗(x) in Eq. (4.62).
- 9.
- 10.
Recall that β and q are the slope and intercept of \(\log \omega ({\bar x})\), therefore an equivalent representation of the ThL transformation is
. - 11.
In this section we notate the MPD as \(f(x;{\bar x})\) to emphasize its dependence on \({\bar x}\).
- 12.
Once the selection functional is fixed, ω, β, and \(\log q\) become immediately fixed. The (a 0, a 1) translation changes the functional and its associated parameters.
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Matsoukas, T. (2018). The Most Probable Distribution in the Continuous Limit. In: Generalized Statistical Thermodynamics. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-030-04149-6_4
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DOI: https://doi.org/10.1007/978-3-030-04149-6_4
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