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This chapter discusses the effects of disorder in fermionic systems, including Anderson localization. There are important differences for the disorder effects between the one-dimensional world, where localization occurs because electrons bump back and forth between impurities, and the higher dimensional world, where Anderson's localization is a rather subtle interference mechanism. The discussion looks at one-dimensional electrons subject to weak and dense impurities, in which the disorder can be replaced by its Gaussian limit. The application of disordered systems to quantum wires, one of the ultimate weapons to study individual one-dimensional systems, is considered.

This chapter starts with an introduction to disordered systems, with an emphasis on disorder in condensed matter, and discusses such phenomena as the Anderson localization, and quantum phases such as with Bose glass. It then turns to various experimental realizations of disorder in ultracold atomic gases. The chapter focuses on disordered Bose-Einstein condensates and the phenomenon of Anderson localization in speckle or quasi-periodic, quasi-random potential. It moves on to disordered ultracold fermionic systems, and disordered ultracold Bose–Fermi (B–F) and Bose–Bose (B–B) mixtures. Finally, the chapter examines spin glasses (providing a brief presentation of the mean field theory of Parisi, and the droplet model) and disorder induced order.

In the presence of disorder, classical transport is usually diffusive. This chapter deals with the effect of interference between multiply scattered waves on transport properties. Interference may lead to reduced diffusive transport—this is known as weak localization—or to complete inhibition of transport—which is known as Anderson localization or strong localization. Key parameters are the dimension of the system and the strength of the disorder. After introductory sections on a transfer-matrix description of 1D transport, scaling theory of localization, and key numerical and experimental results, a general microscopic theory of transport in disordered systems is presented, with emphasis on experimental realizations with cold atomic gases. Simple examples are the propagation of light in a disordered medium, for which we show results from a live coherent backscattering expriment, and the propagation of atomic matter waves in an effective disordered potential created by an optical speckle. Finally, the dynamical localization transition of the kicked rotor, as observed with cold atoms, is discussed.

This chapter reviews current trends in the understanding of the genetics of central nervous system (CNS) disorders and how sex differences impinge on outcomes, or can serve to study the underlying causes of disease. It shows that estrogens appear to play a prominent role, primarily as a protective agent in the case of schizophrenia and depression. Even in depression where women are more strongly affected than men, this may be due to a precipitous drop in estrogens, such as happens after delivering a baby or premenstrually. Yet, hormonal levels are but one of the multiple genomic differences between males and females. The chapter points out the overriding need to consider sex in understanding the disease and optimizing its therapy, but it also highlights the complexity of genomic factors in multigenic disease and therapy.

In the case of spatially limited jump diffusion, the incoherent scattering function consists of an elastic and a quasielastic part, and the fraction of elastic intensity is called the incoherent structure factor, EISF. This chapter evaluates different scenarios and the corresponding EISFs. Long range translational diffusion gives rise to quasielastic scattering consisting of a single Lorentzian with a linewidth proportional to Q². The Chudley–Elliott model for lattice gases describes translational jump diffusion in Bravais lattices. QENS from translational jump diffusion on lattices with all sites equivalent consists of a set of Lorentzians. Even more complex is QENS from diffusion over energetically different sites. The system of rate equation is expressed in terms of a jump matrix, but this matrix is not hermitean. The resulting eigenvalues are the linewidths of the Lorentzians, whereas from the eigenvectors the intensity of the Lorentzians can be calculated. This full power of QENS can only be realized on the basis of direction-dependent measurements on single-crystalline samples. A completely different approach is the so-called two-state model which applies for disordered systems and which is also derived in full detail.

This chapter focuses specifically on ferroic domain walls, the thin interfaces separating different orientations of ferromagnetic magnetization, ferroelectric polarization, or ferroelastic strain in their parent material. Using a variety of optical and local probe microscopy techniques, the static roughening and dynamics of such domain walls can be accessed over multiple orders of length- and timescales, under precisely controlled applied forces (through the use of the appropriate conjugate fields) and at variable temperature, and with the possibility in some cases of tailoring the type and density of defects, making them a powerful model system in which the physics of pinned elastic interfaces can be studied.

This is an introduction to the theory of one-dimensional disordered systems and products of random matrices, confined to the 2×2 case. The notion of impurity model—that is, a system in which the interactions are highly localized—links the two themes and enables their study by elementary mathematical tools. After discussing the spectral theory of some impurity models, Furstenberg’s theorem is stated and illustrated, which gives sufficient conditions for the exponential growth of a product of independent, identically distributed matrices.