We exhibit a simple variable pitch helix model which shows that the “chaotic” behavior of Čebyšev iterations is an illusion of dimensional projection. As a consequence the stroboscopic images of the motion of a simple harmonic oscillator, recorded at the times 2nt (n = 0, 1, 2, …), reproduce the mixing. This power law of time evolution may be used to generate other types of mixing transformations.
The iteration of the Čebyšev polynomial x2 − 2 generates a mixing transformation on the interval x ε [−2, 2]. Extensive computer experiments have demonstrated that this is a convenient method for generating sequences of pseudo-random numbers. Despite the eventual domination of cumulative roundoff errors the asymptotic statistical features of the mixing are preserved. Multiple sequences of stochastically independent variables may be generated by these techniques. In practical computations the Čebyšev mixing eventually terminates in long pseudo-ergodic cycles. These results are linked with the general problem of simulating the stochastic behavior of physical systems by means of functional iteration.
The Fermi surface of beryllium has been throughly investigated by means of the de Haas-van Alphen effect, and the frequencies, though substantially in agreement with earlier measurements, have been obtained to greater accuracy. This has made it possible to construct a nonlocal pseudopotential model for the Fermi surface, in which the Fourier coefficients of the crystal potential are treated as parameters to be evaluated by fitting to the experimental data. In this way, all the principal cross sections of the surface have been fitted to within 1%, using only five adjustable parameters. Comparison of calculated cyclotron masses for these cross sections with the masses determined from the temperature dependence of the dHvA amplitudes indicates that the mass is enhanced about 20% by many-body interactions neglected in the model. Finally, the pressure dependence of the Fermi surface has been calculated and compared with some available experimental measurements.
Recent de Haas-van Alphen data for the Fermi surface of beryllium have been fitted to within 1% using a non-local pseudopotential model.
