We find the general requirements, set by classical electrodynamics, to the sources of extremely high-energy cosmic rays (EHECRs). It is shown that the parameters of EHECR accelerators are strongly limited not only by the particle confinement in large-scale magnetic field or by the difference in electric potentials (generalized Hillas criterion), but also by the synchrotron radiation, the electro-bremsstrahlung, or the curvature radiation of accelerated particles. Optimization of these requirements in terms of accelerator's size and magnetic field strength results in the ultimate lower limit to the overall source energy, which scales as the fifth power of attainable particle energy. Hard γ-rays accompanying generation of EHECRs can be used as a probe for potential acceleration sites. We apply the results to several populations of astrophysical objects -potential EHECR sourcesand discuss their ability to accelerate protons to 10 20 eV and beyond. A possibility to gain from ultrarelativistic bulk flows is emphasized, with Active Galactic Nuclei and Gamma-Ray Bursts being the examples.PACS numbers: 98.70. Sa, 95.30.Gv

We review the phenomenon of equilibrium fluctuations in the number of condensed atoms n0 in a trap containing N atoms total. We start with a history of the Bose-Einstein distribution, a similar grand canonical problem with an indefinite total number of particles, the Einstein-Uhlenbeck debate concerning the rounding of the mean number of condensed atomsn0 near a critical temperature Tc, and a discussion of the relations between statistics of BEC fluctuations in the grand canonical, canonical, and microcanonical ensembles.First, we study BEC fluctuations in the ideal Bose gas in a trap and explain why the grand canonical description goes very wrong for all moments (n0 −n0) m , except of the mean value. We discuss different approaches capable of providing approximate analytical results and physical insight into this very complicated problem. In particular, we describe at length the master equation and canonical-ensemble quasiparticle approaches which give the most accurate and physically transparent picture of the BEC fluctuations. The master equation approach, that perfectly describes even the mesoscopic effects due to the finite number N of the atoms in the trap, is quite similar to the quantum theory of the laser. That is, we calculate a steady-state probability distribution of the number of condensed atoms pn 0 (t = ∞) from a dynamical master equation and thus get the moments of fluctuations. We present analytical formulas for the moments of the ground-state occupation fluctuations in the ideal Bose gas in the harmonic trap and arbitrary power-law traps.In the last part of the review, we include particle interaction via a generalized Bogoliubov formalism and describe condensate fluctuations in the interacting Bose gas. In particular, we show that the canonical-ensemble quasiparticle approach works very well for the interacting gases and find analytical formulas for the characteristic function and all cumulants, i.e., all moments, of the condensate fluctuations. The surprising conclusion is that in most cases the ground-state occupation fluctuations are anomalously large and are not Gaussian even in the thermodynamic limit. We also resolve the Giorgini, Pitaevskii and Stringari (GPS) vs. Idziaszek et al. debate on the variance of the condensate fluctuations in the interacting gas in the thermodynamic limit in favor of GPS. Furthermore, we clarify a crossover between the ideal-gas and weakly-interacting-gas statistics which is governed by a pair-correlation, squeezing mechanism and show how, with an increase of the interaction strength, the fluctuations can now be understood as being essentially 1/2 that of an ideal Bose gas. We also explain the crucial fact that the condensate fluctuations are governed by a singular contribution of the lowest energy quasiparticles. This is a sort of infrared anomaly which is universal for constrained systems below the critical temperature of a second order phase transition.