Scale symmetry renormalization

It turns out that different substances, when subject to phase transition, behave 

The partition function (which all the thermodynamic properties can be computed from) is defined as  

What purpose may such a decimation serve Well, 

This section links together several topics attractors, self-similarity (renormalization group theory), catastrophe theory. 

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We discuss here the basic ideas of the renormalization group, using the discrete chain model. This is not the most elegant or powerful approach, and in Part Til of this book we will present a much more efficient scheme. However, the present approach is conceptually the simplest, and it allows us to explain all the relevant features dilatation symmetry and scaling, fixed points and universality, crossover. Furthermore, technical aspects like the e-expansion also come up. We are then prepared to discuss the Qualitative concept of scaling in its general form and to work out some consequences. 

The finite-size scaling approach to localization assumes that there is a unique function describing the length scaling of the renormalization transformation which is universal for systems of the same symmetry and dimension and that depends only on the Thouless number (22). This function is (3(g), where (for L a small multiple of L)... 

Does the renormalization mean that the Planck scale does not contribute to our experiments No, it does not mean that. The Planck scale indeed contributes, but it does not contribute to the constraint, because it only affects values of masses and charges, however, we do not calculate, but measure them. That makes the Planck-scale effects imobservable. To observe we should compare a measurement and a calculation, but we have only results of measurements. However, there is an option when we should be able to see some effects of the Planck scale [2]. That is a case when we have certain dynamic effects at the Planck scale (e.g., a variation of some constants) or some violation of symmetries which would make our low-energy picture wrong. 

Some models describing phase transitions, particularly in low-dimensional systems, are amenable to exact mathematical solutions. An effective technique for understanding phase transitions is the renormalization group since it can deal with problems involving different length-scales, including the feature of universality, in which very different physical systems behave in the same way neat a phase transition. See also order parameter renormauzation group than-smoN point broken symmetry early universe. 

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