Scaling: The Finite-Size Effect and Crossover Effects

  • Sergey G. AbaimovEmail author
Part of the Springer Series in Synergetics book series (SSSYN)


In the previous chapter, we have considered different approaches to build the renormalization group (RG) transformation. The behavior of a system in the vicinity of its critical point is scale invariant. This allows us to build relationships among different systems of the universality class.

However, in the previous chapter, our primary concern was to study how we can build the RG transformation, and we did not spend much time investigating the emerging scaling.

We overcome this drawback in the current chapter. Initially, we consider the basic principles of scaling (Widom 1965a; Widom 1965b; Domb and Hunter 1965; Patashinski and Pokrovskii 1966; Kadanoff 1966). Then we see that scaling leads to such important concepts as a finite-size effect and crossover effects. Finally, we study the origins of scaling and find that it is described by the formalism of general homogeneous functions. In turn, we demonstrate that the last formalism originates from the scaling hypothesis of the RG transformation.


Correlation Length Percolation Threshold Scaling Function Scaling Parameter Infinite System 
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  1. Abaimov, S.G.: Statistical Physics of Complex Systems (in Russian), 2nd edn. Synergetics: From Past to Future, vol. 57. URSS, Moscow (2013)Google Scholar
  2. Balescu, R.: Equilibrium and Nonequilibrium Statistical Mechanics. Wiley, New York (1975)zbMATHGoogle Scholar
  3. Brankov, J.G.: Introduction to Finite-Size Scaling. Leuven Notes in Mathematical and Theoretical Physics. Series A: Mathematical Physics, vol. 8. Leuven University Press, Leuven (1996)Google Scholar
  4. Cardy, J.: Scaling and Renormalization in Statistical Physics. Cambridge University Press, Cambridge (1996)CrossRefGoogle Scholar
  5. Domb, C., Green, M.S., Domb, C., Lebowitz, J.L. (eds.): Phase Transitions and Critical Phenomena. Academic, London (1972–2001)Google Scholar
  6. Domb, C., Hunter, D.L.: On the critical behavior of ferromagnets. Proc. Phys. Soc. 86(5), 1147 (1965)CrossRefADSGoogle Scholar
  7. Essam, J.W.: Percolation theory. Rep. Prog. Phys. 43(7), 833–912 (1980)CrossRefADSMathSciNetGoogle Scholar
  8. Essam, J.W., Fisher, M.E.: Padé approximant studies of the lattice gas and Ising ferromagnet below the critical point. Chem. Phys. 38(4), 802 (1963)ADSGoogle Scholar
  9. Fisher, M.E.: Rigorous inequalities for critical-point correlation exponents. Phys. Rev. 180(2), 594 (1969)CrossRefADSGoogle Scholar
  10. Goldenfeld, N.: Lectures on Phase Transitions and the Renormalization Group. Perseus, Reading (1992)Google Scholar
  11. Griffiths, R.B.: Ferromagnets and simple fluids near the critical point: Some thermodynamic inequalitites. Chem. Phys. 43(6), 1958 (1965)ADSGoogle Scholar
  12. Griffiths, R.B.: Rigorous results and theorems. In: Domb, C., Green, M.S. (eds.) Exact Results. Phase Transitions and Critical Phenomena, vol. 1, pp. 7–109. Academic, London (1972)Google Scholar
  13. Grimmett, G.R.: Percolation, 2nd edn. Grundlehren der Mathematischen Wissenschaften, vol. 321. Springer, Berlin (1999)Google Scholar
  14. Huang, K.: Statistical Mechanics, 2nd edn. Wiley, New York (1987)zbMATHGoogle Scholar
  15. Josephson, B.D.: Inequality for the specific heat: I. Derivation. Proc. Phys. Soc. 92(2), 269 (1967a)CrossRefADSGoogle Scholar
  16. Josephson, B.D.: Inequality for the specific heat: II. Application to critical phenomena. Proc. Phys. Soc. 92(2), 276 (1967b)CrossRefADSGoogle Scholar
  17. Kadanoff, L.P.: Spin-spin correlation in the two-dimensional Ising model. Nuovo Cim. 44, 276 (1966)CrossRefADSGoogle Scholar
  18. Landau, L.D., Lifshitz, E.M.: Statistical Physics, Part 1, 3rd edn. Course of Theoretical Physics, vol. 5. Pergamon, Oxford (1980)Google Scholar
  19. Ma, S.K.: Modern Theory of Critical Phenomena. Benjamin, Reading (1976)Google Scholar
  20. Patashinskii, A.Z., Pokrovskii, V.L.: Behavior of ordered systems near the transition point. Soviet JETP. 23(2), 292 (1966)ADSGoogle Scholar
  21. Pathria, R.K.: Statistical Mechanics, 2nd edn. Butterworth-Heinemann, Oxford (1996)zbMATHGoogle Scholar
  22. Rushbrooke, G.S.: On the thermodynamics of the critical region for the Ising problem. Chem. Phys. 39(3), 842 (1963)ADSGoogle Scholar
  23. Stanley, H.E.: Introduction to Phase Transitions and Critical Phenomena. Clarendon, Oxford (1971)Google Scholar
  24. Stauffer, D., Aharony, A.: Introduction to Percolation Theory, 2nd edn. Taylor & Francis, London (1994)Google Scholar
  25. Widom, B.: Equation of state in the neighborhood of the critical point. Chem. Phys. 43(11), 3898 (1965a)ADSGoogle Scholar
  26. Widom, B.: Surface tension and molecular correlations near the critical point. Chem. Phys. 43(11), 3892 (1965b)ADSGoogle Scholar

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© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Advanced Structures, Processes and Engineered Materials CenterSkolkovo Institute of Science and TechnologySkolkovoRussia

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