Abstract
If we increase the size of the cluster ensemble at fixed mean, the ensemble becomes a container of every possible cluster distributions n with fixed mean. We call this thermodynamic limit (ThL) and define it by the condition
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Notes
- 1.
We use \(\tilde w_i\) as shorthand for \(w_{i|{\tilde {\mathbf {n}}}}\). As a general notational convention, variables decorated with the tilde, \(\tilde {~}\), refer to the MPD.
- 2.
Recall that the derivatives of \(\log W(\mathbf {n})\) of order k scale as 1∕N k−1 and vanish asymptotically as the size of the system increases. Therefore, all terms of order δ n 2 or higher can be dropped.
- 3.
Two elementary properties of the multinomial distribution
$$\displaystyle \begin{aligned} P(n_1, n_2\cdots) = N! \frac{p_1^{p_1}}{n_1!}\frac{p_2^{n_2}}{n_2!}\cdots \end{aligned}$$are the mean value of n k,
$$\displaystyle \begin{aligned} \bar n_k = N p_k \end{aligned}$$and its variance
$$\displaystyle \begin{aligned} \text{Var}(n_k) = N p_k (1-p_k). \end{aligned}$$ - 4.
As functionals of n, both n! and W(n) are independent of β and are treated as constants in this differentiation.
- 5.
We use the primes in M ′ to distinguish the M ′-canonical subs ensemble from the microcanonical ensemble (M, N) from which it is extracted. We will drop the prime in the thermodynamic limit.
- 6.
Use \(M^{\prime }=\sum _i in^{\prime }_i\), \(N^{\prime }\sum _i n^{\prime }_i\), \(\log W(\mathbf {n^{\prime }})=\sum _i n^{\prime }_i\log \tilde w_i\), \({\tilde p}_i=\tilde w_i e^{-\beta i}/q\), to write
$$\displaystyle \begin{aligned} \mathbf{n^{\prime}!}W(\mathbf{n^{\prime}})\frac{e^{-\beta M^{\prime}}}{q^{N^{\prime}}} = \mathbf{n^{\prime}!}\prod {\tilde p}_i^{n^{\prime}_i} \equiv \text{Prob}\left(\sum_i m_i=M^{\prime}\big|N^{\prime}\right), \end{aligned}$$which is the multinomial probability to sample distribution n ′ in N ′ draws from a pool of clusters with distribution \({\tilde p}_i\). The summation in Eq. (3.49), which can be expressed as
$$\displaystyle \begin{aligned} \sum_{N^{\prime}} \text{Prob}\left(\sum_i m_i=M^{\prime}\big|N^{\prime}\right) , \end{aligned}$$is the probability that the sampled distribution has mass M ′ regardless of the number of clusters.
- 7.
See discussion in Sect. 2.10 on p. 61.
- 8.
Both b and β satisfy the normalization
$$\displaystyle \begin{aligned} \sum_i \tilde w_i e^{-b i} = q, \end{aligned}$$from which we obtain
$$\displaystyle \begin{aligned} \frac{d\log q}{d b} = \frac{d\log q}{d\beta} = -{\bar x}. \end{aligned}$$This implies b = β + c but since both distributions (the one in Eq. (3.56) and the microcanonical MPD) are normalized to unit area, we must have c = 0.
- 9.
More precisely, P(n) converges to a delta function at \(\mathbf {n}={\tilde {\mathbf {n}}}\).
- 10.
In expanded notation this is
$$\displaystyle \begin{aligned} X_i = \sum_{x_1,x_2,\cdots} x_i \, n_{x_1,x_2,\cdots,x_K} \end{aligned}$$with the summation going over all states (x 1, x 2, ⋯ ) in n. Since \(n_{x_1,x_2\cdots }\) is zero for states that are not present in the distribution, we may take the limits of the summation to go from 0 to ∞ for all x i:
$$\displaystyle \begin{aligned} \sum_{x_1,x_2,\cdots} = \sum_{x_1=0}^\infty \sum_{x_2=0}^\infty \cdots . \end{aligned}$$We will abbreviate this as
$$\displaystyle \begin{aligned} \sum_{\mathbf{x}} \end{aligned}$$The same convention applies to the product in Eq. (3.61).
- 11.
- 12.
We can always sort attributes so that the first L of them are those that fluctuate and the remaining K − L are those that are fixed.
- 13.
Using the notation of this section, q and Q in Sect. 3.4 would be notated as q K and Q K, respectively.
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Matsoukas, T. (2018). Thermodynamic Limit (ThL). In: Generalized Statistical Thermodynamics. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-030-04149-6_3
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DOI: https://doi.org/10.1007/978-3-030-04149-6_3
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