Abstract
In the introductory chapter (Chap.1) we showed several examples of critical behaviours near second order phase transitions. Let us remind ourselves of the key observations. When the temperature is near the critical temperature, most physical quantities obey a power law(T−T c )x, where xis called a critical exponent. In this chapter we will see that this sort of behaviour is the signature of the scale invariance of the system close to the critical point.
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Notes
- 1.
Even if we call it f and not g, the quantity we are interested in here is strictly speaking the Gibbs free energy, which we could call the free enthalpy in so far as it depends on the intensive magnetic variable h and not on the corresponding extensive variable, the magnetisation m.
- 2.
For simplicity, the term mean field exponents refers to the exponents calculated within the framework of the mean field approximation.
- 3.
Scaling methods have become a standard tool for physicists today. The word scaling evokes a structure of nature and particular relations, but most of all a universal approach and type of analysis. It refers to an observation as much as to an action, with the idea that the this active approach molds the observation process.
- 4.
Note that more generally the study of changes of scale is also very useful in the case of spatio-temporal phenomena, outside the context of decimation. Take for example a random walk in discrete time (Sect. 4.3.3) obeying the law of diffusion: when we contract time by a factor b, space must be contracted by a factor b α = b 1 ∕ 2 to obtain a statistically identical trajectory. This change of scale factor is the only one for which we obtain a non trivial limit of the diffusion coefficient D at b → ∞. In the case of anomalous diffusion, with “pathological” jump statistics (Lévy flights) or diffusion on a fractal object, the required exponent α is greater than 1 ∕ 2 in the first case (Lévy flights involve superdiffusion) and less than 1 ∕ 2 in the case of a fractal space (where the dead branches lead to subdiffusion).
- 5.
A fixed point \({\mathsf{M}}^{{_\ast}} = ({x}^{{_\ast}},{y}^{{_\ast}},{z}^{{_\ast}},\;\ldots ) = f({x}^{{_\ast}},{y}^{{_\ast}},{z}^{{_\ast}},\;\ldots )\) is said to be stable in the direction x, if f applied iteratively at point \(\mathsf{M} = ({x}^{{_\ast}} + dx,{y}^{{_\ast}},{z}^{{_\ast}},\;\ldots )\) brings M to M ∗ (dx being infinitesimal).
- 6.
It was proved for the renormalisation describing the transition to chaos by doubling the period (see Chap. 9) [Collet and Eckmann 1980].
- 7.
Given that only one of these p choices progresses the line in each direction, the length L takes the value pN 1 ∕ 2 where N is the total number of spins. However, since L is a factor in the expression of f defect, its precise value does not play a role in the sign of f defect.
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Lesne, A., Laguës, M. (2012). Universality as a Consequence of Scale Invariance. In: Scale Invariance. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15123-1_3
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