From order to chaos II : essays, critical, chaotic, and by Leo P Kadanoff

By Leo P Kadanoff

This publication is a compilation of the evaluation papers, expositions and a few of the technical works of Leo Kadanoff, a theoretical physicist. the target is to place jointly a gaggle of not-too-technical writing during which he discusses a few matters in condensed topic physics, hydrodynamics, utilized arithmetic and nationwide policy.This multiplied variation is split into 5 sections. the 1st part comprises evaluation papers on hydrodynamics, condensed topic physics and box thought. subsequent is a range of papers on scaling and universality, really as utilized to part adjustments. a transformation of speed is supplied via a chain of papers at the severe research of simulation versions of city monetary and social improvement. The booklet concludes with a sequence of contemporary papers on advanced styles. every one significant part has an creation designed to tie the interact and to supply viewpoint at the material.

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61b), represents the contribution of the semi-infinite part of the chain at the right side of the site under attention, while the other fractions represent the contribution of one or two sites at the left of the site under attention. 62c), a simple calculation gives (s) (s) n 1 (E) = n 0 (E) E2 E2 (s) (s) , n (E) = n (E) −1 2 0 γ2 γ2 2 . 15. 51); we only mention that several significant properties such as localization effects in incommensurate systems, dimerized systems, disordered systems etc.

27). The transmission coefficient of the double barrier is then T (tot) = 1 (tot) |m 11 |2 =⇒ T (tot) (E) = T 2 (E) 1 + R(E)e−2i(φt +qb+qw) 2 . 36) From the above relation, it is evident that the transmission coefficient of the double barrier equals unity (perfect transparency) when the transmission coefficient T of the single well equals unity (and R = 0); resonance energies of the single barrier are resonance energies for the double barrier, too. More importantly, it is seen by inspection that perfect transparency also occurs when e−2i(φt +qb+qw) ≡ −1 or equivalently cos (φt + qb + qw) ≡ 0.

Notice that all the degeneracies of the empty lattice are fully removed (even if the direct coupling between degenerate wavefunctions of the empty lattice occurs only for k = ±π/a) [for further aspects of the Mathieu problem see J. C. Slater, Phys. Rev. 87, 807 (1952)]. The Mathieu potential has been originally introduced as a model to study and mimic some features of the physics of electronic states of crystalline materials; however, the ideal experimental realization of systems described by the Mathieu model has occurred several decades later in the physics of ultracold atoms in optical lattices, where atoms of mass M (for instance sodium or rubidium) are trapped in a standing wave of laser light with wavelength λ ≈ 104 Å.

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