Abstract
The generic population that is the subject of our study consists of M indistinguishable members assembled into N groups, or clusters, such that no group is empty. The “member” is the fundamental unit of the population and plays the same role as “monomer” in a polymeric system, or “primary particle” in granular materials. The cluster is characterized by the number of members it contains. We will refer to the number of members in the cluster as the size or mass of the cluster and will use the terms interchangeably. The goal in this chapter is to define a sample space of distributions of clusters and assign a probability measure over it. This probability space of distributions will be called cluster ensemble and forms the basis for the development of generalized thermodynamics. In Chap. 7 we will reformulate the theory on the basis of a more abstract space of distributions.
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- 1.
We are assuming that the population consists of a single component. We will later generalize to clusters made of any number of distinguishable components.
- 2.
See Bóna (2006), p. 90.
- 3.
- 4.
We use the notation w i as opposed to w i|n to indicate that w i is the same for all n.
- 5.
Since moments are functionals of the distribution, it is perfectly acceptable to have selection functionals that reduce to pure functions of certain moments. What makes N and M special is that these are the only two moments that evaluate to the same number in every distribution of the ensemble.
- 6.
- 7.
It is sufficient to work with a die that has M − N + 1 sides, since this is the maximum possible number in N tosses that add up to M.
- 8.
We will revisit this idea in Chap. 7.
- 9.
We previously used p i for the cluster mass probability. Here we use P(i|n) to emphasize that this probability refers to specific distribution n.
- 10.
As previously, we are using the notation W(m) to refer to the bias of configuration m; this is equal to W(n), where n is the cluster distribution in m.
- 11.
For example, perform a series of transfers to a single cluster until its mass is M − N + 1 (and all other clusters are monomers), then transfer from that cluster to the rest until the desired configuration is reached.
- 12.
- 13.
In writing ΩM−jk,N−j we understand the subscripts to satisfy the conditions M − jk ≥ N − j ≥ 1.
- 14.
A mass m in a given column of the microcanonical table is paired with every permutation of every configuration that contains N − 1 clusters with total mass M − m; their number is the volume of the (M − m, N − 1) ensemble.
- 15.
In the microcanonical ensemble the largest and smallest masses are M − N + 1 and 1, respectively; their difference is M − N.
- 16.
We must also have N′≤ N − 1 because the complement must contain at least one cluster. Therefore, the condition in (2.69) applies to all slices except the last one. The last column will be obtained by mass balance.
- 17.
A note on nomenclature: Here we use n ′ to indicate the canonical distribution, n″ to indicate its microcanonical complement, and n = n ′ + n″ to indicate the microcanonical distribution that is formed by combining the canonical distribution with its complement.
- 18.
We may rename the canonical ensemble to N′-canonical, to be consistent with the nomenclature used in this section. We choose not do this and will continue to refer to the N′-canonical ensemble simply as canonical. We will use the term N′-canonical only where needed to avoid possible confusion when discussing both canonical ensembles together.
- 19.
The fraction of M′-canonical configurations with M′ = 5 in this example is 20∕35 = 0.57, fairly close to N∕M = 5∕8 = 0.625.
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Matsoukas, T. (2018). The Cluster Ensemble. In: Generalized Statistical Thermodynamics. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-030-04149-6_2
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