By Stephan I. Tzenov
This booklet adopts a non-traditional method of accelerator idea. The exposition starts off with the synchro-betatron formalism and maintains with the linear and nonlinear theories of transverse betatron movement. a variety of equipment of learning nonlinear dynamical structures (the canonical conception of perturbations and the tools of a number of scales and formal sequence) are defined via examples. The renormalization staff method of learning nonlinear (continuous and discrete) dynamical platforms as utilized to accelerators and garage earrings is used through the publication. The statistical description of charged particle beams (the Balescu–Lenard and Landau kinetic equations in addition to the Vlasov equation) is handled within the moment a part of the publication. The strategies of trend formation and formation of coherent constructions (solitons) also are defined.
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8) corresponds to treating the dynamics of the condensate as if it were moving in a static mean ﬁeld of the noncondensate thermal cloud. One then obtains i¯ h h2 ∇2 ¯ ∂Φ n0 (r) + gnc (r, t) Φ(r, t). 10) There is a considerable literature based on the static thermal cloud in the Popov approximation (m(r, ˜ t) = 0). 10). The self-consistent calculation of these two densities determines the equilibrium properties of the trapped gas within the static Popov approximation. The excitation energies are the quantized ﬂuctuations of the condensate.
1 Generalized GP equation for the condensate Page-33 33 leads to the damping (or growth) of condensate ﬂuctuations. The C12 collisions will play a crucial role in the rest of this book. 2 follows the original approach given by Zaremba et al. (1999). Throughout this chapter (and the whole book), this paper will be referred to as ZNG. The derivation of the collision integrals given in Appendix A of ZNG follows the approach of Kirkpatrick and Dorfman (1985a). We will not repeat this derivation here since the same results are derived in Chapter 6 using the more transparent and systematic Kadanoﬀ–Baym formalism.
Apart from this application, the GP equation was largely unknown. The situation changed overnight in 1995 with the creation of trapped nonuniform Bose condensates in atomic gases. This chapter is a review of the ground state solution of the T = 0 GP equation and of small-amplitude ﬂuctuations (collective oscillations) about this equilibrium state. This review is needed as the starting point for generalizations in the following chapters, which deal with ﬁnite temperatures. 19 BECBook CUP/GFN 20 November 4, 2008 16:42 Page-20 Condensate dynamics at T = 0 For much more detailed accounts of the GP equation at T = 0, we refer to the review by Fetter (1999) as well as the texts by Pethick and Smith (2008) and Pitaevskii and Stringari (2003).