![]() This angle is completely independent of the incoming angle. Recent experiments have shown that many species of microorganisms leave a solid surface at a fixed angle determined by steric interactions and near-field hydrodynamics. Microorganism billiards in closed plane curves. This can be understood as an indication that quantum Darwinism is present in the electron billiards. they can be observed in the individual modes propagating between the external reservoirs. Additionally, we show that the two types of pointer states have the propensity to create offspring, i. We report on the existence of pointer states in single-dot and double-dot electron billiards. In this paper, we investigate the dynamics in electron billiards by using classical and quantum mechanical calculations. Indication for quantum Darwinism in electron billiardsīrunner, R. It is shown, in particular, that the second moment of scattering about the incidence angle closely approximates the spectral gap of P. 2), that the billiard Laplacian P- I is closely related to the ordinary spherical Laplacian, and indicate, by partly analytical and partly numerical means, how this provides asymptotic information about the spectrum of P for small values of K. We show, for a particular family of billiard cell shapes parametrized by a scale invariant curvature K (Fig. A central problem in the statistical theory of such random billiards is to relate the geometric characteristics of Q and the spectrum of P. This operator, defined on an appropriate Hilbert space, is bounded self-adjoint and, for the examples considered here, a Hilbert-Schmidt operator. The microstructure in turn is defined in terms of what we call a billiard cellQ, the shape of which completely determines the operator P. Billiards with microstructure are random billiards whose Markov operator is derived from a "microscopic surface structure" on the boundary of the billiard table. Random billiards are billiard dynamical systems for which the reflection law giving the post-collision direction of a billiard particle as a function of the pre-collision direction is specified by a Markov (scattering) operator P. The Spectrum of the Billiard Laplacian of a Family of Random Billiards We apply this theory to find the dynamics of the IIA and IIB strings, the M2 and M5 branes, the IIB D3Â brane as well as the one and two branes in seven dimensions. Each brane carries the full E11 symmetry and so the Cremmer-Julia duality symmetries. The resulting equations of motion are first order in derivatives and can be thought of as duality relations. The brane moves through a space-time which arises in the nonlinear realisation from the vector representation and it contains the usual embedding coordinates as well as the worldvolume fields. E11, brane dynamics and duality symmetriesįollowing arXiv:hep-th/0412336 we use the nonlinear realisation of the semi-direct product of E11 and its vector representation to construct brane dynamics.
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