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This chapter discusses the quantum phase transition using the example of superconductor-insulator transition. For the 3D case, a version of the Ginzburg-Landau formalism is formulated from both normal and anomalous diffusion. The description of such transition in the case of 2D superconductors is very specific and strongly differs from the 3D case. The ideas of boson and fermion mechanisms of the T_c suppression in 2D cases are presented, and their predictions are compared with the experimental conditions.

Tsallis generalized statistics has been successfully applied to describe some relevant features of several natural systems exhibiting a nonextensive character. It is based on an extended form for the entropy, namely [math], where q is a parameter that measures the degree of nonextensivity (q → 1 for the traditional Boltzmann-Gibbs statistics). A series of recent works have shown that the power-law sensitivity to initial conditions in a complex state provides a natural link between the q-entropic parameter and the scaling properties of dynamical attractors. These results contribute to the growing set of theoretical and experimental evidences that Tsallis statistics can be a natural frame for studying systems with a fractal-like structure in phase space. Here, the main ideas underlying this relevant aspect are reviewed. The starting point is the weak sensitivity to initial conditions exhibited by low-dimensional dynamical systems at the onset of chaos. It is shown how general scaling arguments can provide a direct relation between the entropic index q and the scaling exponents associated with the multifractal critical attractor.

These works shed light in the elusive problem concerning the connection between the q-entropic parameter of Tsallis statistics and the underlying microscopic dynamics of nonextensive systems.

The renormalization-group (RG) method discussed in this chapter has assumed a pivotal role in the modern theory of critical phenomena. It attempts to relate the partition function of a given system to that of a “similar system” with decreased degrees of freedom through a process referred to as renormalization. Exactly how these degrees of freedom are removed from the system, what we mean by a “similar” system, and how successive systems are coupled to one another are essentially the questions we take up in the introductory treatment given in this chapter. The RG method is a topic with large scope and found widely disseminated in an extensive physics literature on the topic; however, it is seldom found in engineering journals. Our purpose here is to try and make sense of some basic ideas with the RG approach so that it is more accessible to this wider community. For this we often rely upon some prior exposes of the subject in more specialized settings [1, 2, 3, 5]. In its complete sense, the RG method has only been made to work, at least analytically, for a few simple statistical-mechanical models. But aside from these numerical results, many important and quite general insights about critical phenomena can be developed from studying this approach to the problem, especially the central role played by length scale as a factor in describing the phenomenology. These ideas have significantly enhanced our understanding of ideas like scale invariance, universality classes, relevant scaling fields (as opposed to irrelevant ones), Hamiltonian renormalization, and so on; these and related concepts lie at the center of modern discourse on the subject. The essential concepts of the approach can be well illustrated using the Ising system since, with this model, lattice spins are fixed in space, which makes the analytical work quite transparent. This approach, called real space renormalization, is the RG method studied in this chapter.