Abstract
In the previous chapter, we have considered the phase transition phenomenon in the Ising model. May we call this system “complex?” In the literature there is no settled definition of what we call “complex.”
The theory of phase transitions is generally attributed to thermodynamic systems. However, in the second half of the last century many nonthermal systems have been discovered whose behavior resembled the theory of phase transitions in statistical physics. However, these systems belong to such diverse sciences—biology, geology, engineering, chemistry, mathematics, economics, social sciences, etc.—that their unified classification seems to be impossible. Examples include but not limited to the percolation of petroleum oil in a rock, polymerization, damage of engineering structures, earthquakes, forest fires, snow avalanches and landslides, traffic jams, chaotic systems, strange attractors, informational processes, self-organized criticality, etc.
To distinguish such systems from the classical examples of phase transition phenomena (like the Ising model), the term “complex” has appeared. However, beyond the fact that all these systems obey the rules of phase transitions and, therefore, can form universality classes, their common rigorous classification is deemed to be currently impossible.
We see that the term “complex” is collective and, therefore, may describe a great variety of phenomena. But what does this term mean? First, as we have said, calling a system complex, one generally assumes that this complexity is the consequence of a phase transition (or a bifurcation, catastrophe, nonanalyticity, etc.) present in the system. Second, the term “complex” is generally used to distinguish the nonthermal systems from their thermodynamic analogues.
Summarizing, we call a system complex if it possesses a phase transition but is nonthermal. In the sense of this definition, the Ising model is only partially complex—it possesses a phase transition but is thermal. In this chapter, as a first example of a “completely complex” system, we consider a phenomenon of percolation.
The fact that the system is supposed to be nonthermal means that fluctuating behavior is no longer described by thermodynamic fluctuations. Instead, the system must possess another source of stochastic behavior, forming nonthermal fluctuations.
Once the nonthermal fluctuations are generated, our main purpose is to map these fluctuations on their thermodynamic analogs so that the well-developed formalism of the theory of phase transitions in statistical physics may become available for their description.
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Notes
- 1.
This equation resembles very much the law of conservation of particles in the case of the Bose–Einstein condensation. In both cases, after the critical point, one term of the discret sum begins to represent the number of degrees of freedom which is comparable with N. So, in the case of the Bose–Einstein condensation, the number of particles in the condensate becomes comparable with the total number of particles in the system while in percolation the number NP PC (p) of sites belonging on average to a percolating cluster occupies a significant part of the lattice. In both cases, the considered term is separated from the sum to emphasize its outstanding role and not to lose it when the sum is substituted by an integral.
- 2.
While the statement is valid, the reasoning behind it is oversimplified. Further we apply the logarithmic accuracy in the limit of big clusters: \(s>> 1/c\) . To find the probability for a cluster perimeter to be smooth instead of being more complex (fractal) we should compare the part of the normalized cluster number that comes from the lattice animals with smooth perimeters, \(\Delta {n_s} = \underline{\underline {\rm O}} ({s^\alpha }){p^s}{(1 - p)^{A{s^{\frac{{d - 1}}{d}}}}}{ \approx _{\ln }}{p^s}{(1 - p)^{A{s^{\frac{{d - 1}}{d}}}}}\) with the share of the normalized clus-ter number belonging to the complex lattice animals, \(\Delta {n_s}{ \approx _{\ln }}{e^{Cs}}{p^s}{(1 - p)^{Bs}}\), where we have taken into account that the numbers \({g_{{t_s}}}\) of lattice animals with perimeters \({{t}_{s}}\) are the power-law dependences \(\underline{\underline {\rm O}} ({s^\alpha })\) on s for \({t_s} \propto A{s^{\frac{{d - 1}}{d}}}\) and the exponential dependences \({e^{Cs}}\) on s for \({t_s} \propto Bs\). In the limit \(p \to 1 - 0\), comparing \(A{s^{\frac{{d - 1}}{d}}}\ln (1 - p)\) with \(s\left\{ {C + B\ln (1 - p)} \right\} \propto Bs\ln (1 - p)\), we observe that the first ex-pression has smaller absolute value but is higher than the second expression when we take into account the negative sign: \(s\left\{ {\,... < 0\,} \right\} < < {s^{\frac{{d - 1}}{d}}}\left\{ {\,... < 0\,} \right\}\). In other words, the number of complex lattice animals does be much bigger than the number of animals with smooth perimeters, but the improbability to have larger perimeters cancels this advantage, leaving the leading role to smooth, non-fractal, compact clusters, \({n_s}{ \approx _{\ln }}{p^s}{(1 - p)^{A{s^{\frac{{d - 1}}{d}}}}}\), which proves the statement above.
- 3.
The argument in the footnote on the previous page is no longer valid when we consider p to be well below unity. In this case \(C + B\ln (1 - p)\) can become positive, transferring the leading role to complex clusters, \({n_s}{ \approx _{\ln }}{e^{Cs}}{p^s}{(1 - p)^{Bs}}\), while the compact clusters are neglected now. The transition occurs at \({p_0} = 1 - {e^{ - C/B}}\), where C can be estimated with the aid of the total number g s of lattice animals with s sites. Indeed, neglecting the number of compact lattice animals in comparison with their more complex counterparts, \({g_s} = \underline{\underline {\rm O}} ({s^\alpha }) + {e^{Cs}}{ \approx _{\ln }}{e^{Cs}}\), we find \(C = \ln {g_s}/s\). Thereby, for \({p_0} < p < 1\) the normalized cluster number represents the behav-ior of smooth perimeter clusters: \({n_s}{ \approx _{\ln }}{p^s}{(1 - p)^{A{s^{\frac{{d - 1}}{d}}}}}\). On the contrary, for \({p_C} < p < {p_0}\) the cluster-size distribution is determined by complex clusters with: \({t_s} \propto Bs:{n_s}{ \approx _{\ln }}{e^{Cs}}{p^s}{(1 - p)^{Bs}}\). The constant B here we can estimate from the requirement that \({{n}_{s}}\) has a maximum in the vicinity of the percolation threshold: \(0 \approx {\left. {\frac{{d\ln {n_s}}}{{dp}}} \right|_{{p_C}}} = \frac{s}{{{p_C}}} - \frac{{Bs}}{{1 - {p_C}}}\); or \(B \approx \frac{{1 - {p_C}}}{{{p_C}}}\). On square lattice \({e^C} \approx 4.06\), \({p_C} \approx 0.593\) and, thereby, \(B \approx 0.686\). Therefore, for p 0 our argument provides \({p_0} \approx 0.870\) which is indeed greater than p c.
References
Abaimov, S.G.: Statistical Physics of Complex Systems, 2nd ed. Synergetics: From Past to Future, vol. 57, URSS, Moscow (2013) (in Russian)
Bollobás, B., Riordan, O.: Percolation. Cambridge University Press, Cambridge (2006)
Broadbent, S.R., Hammersley, J.M.: Percolation processes. I. Crystals and mazes. Math. Proc. Camb. Phil. Soc. 53(3), 629–641 (1957)
Bunde, A., Havlin, S. (eds.): Fractals and Disordered Systems, 2nd ed. Springer, New York (1996)
Deutscher, G., Zallen, R., Adler, J. (eds.): Percolation Structures and Processes. Adam Hilger: Bristol (1983)
Essam, J.W., Gwilym, K.M.: The scaling laws for percolation processes. J. Phys. C 4(10), L228–L231 (1971)
Feder, J.: Fractals. Physics of Solids and Liquids. Springer, New York (1988)
Fisher, M.E.: The theory of condensation and the critical point. Physics (Long Island City, NY) 3, 255–283 (1967a)
Fisher, M.E.: The theory of equilibrium critical phenomena. Rep. Prog. Phys. 30(2), 615 (1967b)Fisher, M.E., Essam, J.W.: Some cluster size and percolation problems. J. Math. Phys. 2(4), 609–619 (1961)
Flory, P.J.: Molecular size distribution in three dimensional polymers. I. Gelation. J. Am. Chem. Soc. 63(11), 3083–3090 (1941a)
Flory, P.J.: Molecular size distribution in three dimensional polymers. II. Trifunctional branching units. J. Am. Chem. Soc. 63(11), 3091–3096 (1941b)
Flory, P.J.: Molecular size distribution in three dimensional polymers. III. Tetrafunctional branching units. J. Am. Chem. Soc. 63(11), 3096–3100 (1941c)
Grimmett, G.R.: Percolation. Grundlehren der Mathematischen Wissenschaften, 2nd ed, vol. 321. Springer, Berlin (1999)
Kesten, H.: Percolation Theory for Mathematicians. Progress in Probability and Statistics, vol. 2. Birkhäuser, Boston (1982)
Sahimi, M.: Applications of Percolation Theory. Taylor & Francis, London (1994)
Stauffer, D., Aharony, A.: Introduction to Percolation Theory, 2nd ed. Taylor & Francis, London (1994)
Stockmayer, W.H.: Theory of molecular size distribution and gel formation in branched-chain polymers. J. Chem. Phys. 11(2), 45–55 (1943)
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Abaimov, S. (2015). The Theory of Percolation. In: Statistical Physics of Non-Thermal Phase Transitions. Springer Series in Synergetics. Springer, Cham. https://doi.org/10.1007/978-3-319-12469-8_4
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