Abstract
The simplest cluster ensemble is formed by partitioning an extensive variable M, which we have called “mass,” into N clusters. In Chap. 3 we generalized this to the partitioning of a set of extensive variables, X 1, X 2, ⋯ into N clusters. X i may represent energy, volume, or any other extensive attribute that is distributed. A special case is when this attribute refers to a distinct species that we recognize as a component. A population that contains two or more components forms a mixture and its behavior is quite different from that of the generic multivariate ensemble. All extensive properties of a multicomponent population may be sub-partitioned with respect to components. In this chapter we formulate the bicomponent cluster ensemble, derive its thermodynamics, and study the mixing of components for certain classes of selection functionals.
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- 1.
The reference state is a rule that allows us to factor the microcanonical weight into two factors, one that we call multiplicity, and one that we call selection functional. When we use the term “natural multiplicity,” we are in fact referring to the multiplicity given the chosen reference state. “Natural” in this context should not be taken to imply a universal quality.
- 2.
We make use of the identities,
$$\displaystyle \begin{aligned} \sum_{a=0}^k\binom{k}{a} c_1^a (1-c_1)^{k-a} = 1,\quad \sum_{a=0}^k\binom{k}{a} a c_1^a (1-c_1)^{k-a} = k c_1. \end{aligned}$$From the latter identity and Eqs. (6.72) and (6.69) we obtain
$$\displaystyle \begin{aligned} kc_1 = \frac{M_A}{N_k} = \frac{k M_A}{M_A+M_B} = k \phi_A, \end{aligned}$$and finally c 1 = ϕ A.
- 3.
“Ideality” is a dated term still used in thermodynamics to refer to a convenient reference state (several such states are in common use, including ideal gas, ideal solution, and ideal solute). The preferred term is entropy of random mixing, which states the reference state explicitly.
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Matsoukas, T. (2018). The Bicomponent Ensemble. In: Generalized Statistical Thermodynamics. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-030-04149-6_6
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DOI: https://doi.org/10.1007/978-3-030-04149-6_6
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