Abstract
A problem of great theoretical interest in thermodynamics is the possibility of phase transitions. Many dynamical systems exhibit behavior that is remarkably similar to phase transitions. Polymer gelation, shattering in fragmentation, the spread of epidemics, and the emergence of long-range connectivity in artificial and neural networks are examples of the emergence of a giant coherent structure, a behavior that is often discussed qualitatively in the language of phase transitions. If generic population obey thermodynamics, do they also undergo phase transitions? The answer is, yes. As in molecular thermodynamics, phase splitting in the cluster ensemble is associated with the violation of the stability conditions that guarantee the existence of a maximum in the microcanonical weight that defines the most probable distribution. In this chapter we formalize the stability conditions that ensure the existence of the most probable distribution and identify the giant cluster as a phase that is distinct from the sol, a stable population of dispersed clusters. We discuss two mathematical models that give rise to a giant cluster and solve them analytically. A kinetic model of gelation with a closer connection to a physical system of reacting polymers will be discussed in Chap. 9.
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Notes
- 1.
Apply the homogeneity condition in the form \(\log \Omega _{M,N}= N\log \Omega _{M/N,1}\) and notice that
$$\displaystyle \begin{aligned} \log\Omega_{M/N,1} = \frac{\log\Omega_{M,N}}{N} = \log\omega({\bar x}) . \end{aligned}$$ - 2.
Let F(x, y) = xf(x∕y), which makes F homogeneous in x and y with degree 1. The second derivatives of F are related to those of f as follows:
$$\displaystyle \begin{aligned} F_{xx} = \frac{y^2}{x^3}f''(y/x),\quad F_{yy} = \frac{1}{x}f''(y/x) . \end{aligned}$$If x and y are both positive, F has the same curvature (convex or concave) with respect to both variables as f. The relationship between f and F is the same as between \(\log \omega \) and \(\log \Omega \) and since \(\log \omega \) is concave, so is \(\log \Omega \).
- 3.
Both \({\bar x}\) and q are positive.
- 4.
The advantage in recycling terminology is that the theory may be more transparent and accessible to those familiar with standard thermodynamics. The potential downside is that this familiarity may carry with it preconceived notions about these terms that conflict with our definitions.
- 5.
The statement “the MPD maximizes the partition function” is technically incorrect. The partition function is a function of M, N, not of the MPD, and cannot be maximized by the MPD. What we mean is that the MPD maximizes the microcanonical weight n! W(n), whose maximum value is equal to the partition function. This should be understood as implied when we use the shorthand “MPD maximizes Ω.”
- 6.
See, for example, Spencer (2010).
- 7.
In the multivariate ensemble (M 1, M 2⋯ ; N) Eq. (5.18) applies to all β i. If we make the associations
$$\displaystyle \begin{aligned} M_1\to\text{energy},\, M_2\to\text{volume},\, M_{2+i}\to\text{number of particles of component {$i$}},\end{aligned} $$then
$$\displaystyle \begin{aligned} \beta_1=1/k_B T,\; \beta_2=p/k_B T,\; \beta_{2+i}=\mu_i/k_B T,\end{aligned} $$and we recover the equality of temperature, pressure, and all chemical potentials in every part as the condition of equilibrium. We defer the discussion of the connection between the cluster ensemble and statistical mechanics until Chap. 7.
- 8.
This can be confirmed by expanding the scale in Fig. 5.4. We choose a narrow range in the vertical axis to highlight the minimum.
- 9.
If \({\bar x}\) is of the order \(1/\sqrt {2|\gamma |}\) the distribution is a truncated Gaussian function and can be calculated numerically.
- 10.
The minimum gel size is
$$\displaystyle \begin{aligned} i_{\text{gel}} = \frac{M-N+2}{2} = \frac{M\theta}{2} + \frac{2}{M} . \end{aligned}$$This is plotted as dotted line in Fig. 5.17.
- 11.
There is a discontinuity between the fraction of the excess mass right before the gel point and the gel fraction right after it, though of smaller magnitude than the discontinuity in ϕ gel. It arises from the fact that β and q of the equilibrium sol vary abruptly in the region marked CD in Fig. 5.15.
References
J. Spencer, The giant component: The golden anniversary. Not. AMS 57(6), 720–724 (2010)
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Matsoukas, T. (2018). Phase Transitions: The Giant Cluster. In: Generalized Statistical Thermodynamics. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-030-04149-6_5
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DOI: https://doi.org/10.1007/978-3-030-04149-6_5
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