Scaling: The Finite-Size Effect and Crossover Effects
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.
KeywordsCorrelation Length Percolation Threshold Scaling Function Scaling Parameter Infinite System
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