The Renormalization Group
In the previous chapters, we saw that the mean-field approach always determines the critical or spinodal indices as simple integers or rational fractions, like 1 or 5/2. Even more, such indices are considered to be an indicator that the behavior of a system is dominated by the mean-field approach. If the exact solution of a problem provided such simple numbers, the dimensionality of the considered system would probably be above the upper critical dimension, when the mean-field approach represents the exact critical indices, or interactions in the system would be long-range which would lead to the same result.
However, as we know from Chap. 6, the Ginzburg criterion states that if the dimensionality of a system with short-range interactions is lower than the upper critical dimension (which generally corresponds to our three-dimensional space), the mean-field approach is too crude to describe the behavior of the system within the critical region. The mean-field approach may still be considered as a good illustration of a phase transition, but the predicted values of the critical indices are far from being accurate.
Besides, as we discussed in Chap. 6, the mean-field approach is not at all capable to explain the influence of the dimensionality of a system on its behavior—there would be much poorer diversity of the critical indices if all systems obeyed the mean-field approach exactly.
And what is even worse, it is not possible to improve the accuracy of the mean-field approach within the method itself. Only the introduction of newer approaches, within the mean-field approach as well as independent, can improve the situation.
In this chapter, we consider the renormalization group (RG) of coarse graining as an alternative approach to the mean-field approximation. The critical indices determined by this technique are no longer simple integers or fractions. And what is more important is that the RG approach contains, within its formalism, the recipes of how to make calculations more accurate so that the predicted results would be closer to the experimental values.
That is why the RG has been met with such general approval in the literature. Even for the critical indices themselves, to distinguish them from the “habitual” mean-field values, a special term anomalous dimension has been introduced which represents the difference between the real index and its value provided by Landau theory.
KeywordsPartition Function Renormalization Group Correlation Length Flow Curve Percolation Threshold
- Abaimov, S.G.: Statistical Physics of Complex Systems, 2nd ed. Synergetics: From Past to Future, vol. 57, URSS, Moscow (2013) (in Russian)Google Scholar
- Domb, C., Green, M.S., Domb, C., Lebowitz, J.L. (eds.): Phase Transitions and Critical Phenomena. Academic Press, London (1972–2001)Google Scholar
- Goldenfeld, N.: Lectures on Phase Transitions and the Renormalization Group. Perseus Books Publishing, Reading (1992)Google Scholar
- Grimmett, G.R.: Percolation, 2nd ed. Grundlehren der Mathematischen Wissenschaften, vol. 321. Springer, Berlin (1999)Google Scholar
- Kadanoff, L.P.: Scaling laws for Ising models near T C. Phys. (Long Island City, NY) 2, 263 (1966)Google Scholar
- Landau, L.D., Lifshitz, E.M.: Statistical Physics, Part 1, 3rd ed. Course of Theoretical Physics, vol. 5. Pergamon Press, Oxford (1980)Google Scholar
- Ma, S.K.: Modern Theory of Critical Phenomena. Benjamin, Reading (1976)Google Scholar
- Newman, W.I., Gabrielov, A.M.: Failure of hierarchical distributions of fiber-bundles. I. Int. J. Fract. 50(1), 1–14 (1991)Google Scholar
- Niemeijer, T., van Leeuwen, J.M.J.: Renormalization theory for Ising-like spin systems. In: Domb, C., Green, M.S. (eds.) Phase Transitions and Critical Phenomena, vol. 6, pp. 425–505. Academic Press, London (1976)Google Scholar
- Sornette, D., Sammis, C.G.: Complex critical exponents from renormalization-group theory of earthquakes: Implications for earthquake predictions. J. Phys. I 5(5), 607–619 (1995)Google Scholar
- Stanley, H.E.: Introduction to Phase Transitions and Critical Phenomena. Clarendon Press, Oxford (1971)Google Scholar
- Stauffer, D., Aharony, A.: Introduction to Percolation Theory, 2nd ed. Taylor & Francis, London (1994)Google Scholar