Recommendations and illustrations for the evaluation of photonic random number generators

The never-ending quest to improve the security of digital information combined with recent improvements in hardware technology has caused the field of random number generation to undergo a fundamental shift from relying solely on pseudo-random algorithms to employing optical entropy sources. Despite these significant advances on the hardware side, commonly used statistical measures and evaluation practices remain ill-suited to understand or quantify the optical entropy that underlies physical random number generation. We review the state of the art in the evaluation of optical random number generation and recommend a new paradigm: quantifying entropy generation and understanding the physical limits of the optical sources of randomness. In order to do this, we advocate for the separation of the physical entropy source from deterministic post-processing in the evaluation of random number generators and for the explicit consideration of the impact of the measurement and digitization process on the rate of entropy production. We present the Cohen-Procaccia estimate of the entropy rate $h(\epsilon,\tau)$ as one way to do this. In order to provide an illustration of our recommendations, we apply the Cohen-Procaccia estimate as well as the entropy estimates from the new NIST draft standards for physical random number generators to evaluate and compare three common optical entropy sources: single photon time-of-arrival detection, chaotic lasers, and amplified spontaneous emission

On the streaming model for redshift-space distortions

The streaming model describes the mapping between real and redshift space for 2-point clustering statistics. Its key element is the probability density function (PDF) of line-of-sight pairwise peculiar velocities. Following a kinetic-theory approach, we derive the fundamental equations of the streaming model for ordered and unordered pairs. In the first case, we recover the classic equation while we demonstrate that modifications are necessary for unordered pairs. We then discuss several statistical properties of the pairwise velocities for DM particles and haloes by using a suite of high-resolution $N$-body simulations. We test the often used Gaussian ansatz for the PDF of pairwise velocities and discuss its limitations. Finally, we introduce a mixture of Gaussians which is known in statistics as the generalised hyperbolic distribution and show that it provides an accurate fit to the PDF. Once inserted in the streaming equation, the fit yields an excellent description of redshift-space correlations at all scales that vastly outperforms the Gaussian and exponential approximations. Using a principal-component analysis, we reduce the complexity of our model for large redshift-space separations. Our results increase the robustness of studies of anisotropic galaxy clustering and are useful for extending them towards smaller scales in order to test theories of gravity and interacting dark-energy models.Comment: 22 pages, 20 figures, accepted for publication in MNRA

Packet Loss in Terrestrial Wireless and Hybrid Networks

The presence of both a geostationary satellite link and a terrestrial local wireless link on the same path of a given network connection is becoming increasingly common, thanks to the popularity of the IEEE 802.11 protocol. The most common situation where a hybrid network comes into play is having a Wi-Fi link at the network edge and the satellite link somewhere in the network core. Example of scenarios where this can happen are ships or airplanes where Internet connection on board is provided through a Wi-Fi access point and a satellite link with a geostationary satellite; a small office located in remote or isolated area without cabled Internet access; a rescue team using a mobile ad hoc Wi-Fi network connected to the Internet or to a command centre through a mobile gateway using a satellite link. The serialisation of terrestrial and satellite wireless links is problematic from the point of view of a number of applications, be they based on video streaming, interactive audio or TCP. The reason is the combination of high latency, caused by the geostationary satellite link, and frequent, correlated packet losses caused by the local wireless terrestrial link. In fact, GEO satellites are placed in equatorial orbit at 36,000 km altitude, which takes the radio signal about 250 ms to travel up and down. Satellite systems exhibit low packet loss most of the time, with typical project constraints of 10−8 bit error rate 99% of the time, which translates into a packet error rate of 10−4, except for a few days a year. Wi-Fi links, on the other hand, have quite different characteristics. While the delay introduced by the MAC level is in the order of the milliseconds, and is consequently too small to affect most applications, its packet loss characteristics are generally far from negligible. In fact, multipath fading, interference and collisions affect most environments, causing correlated packet losses: this means that often more than one packet at a time is lost for a single fading even

Mining for cosmological information: Simulation-based methods for Redshift Space Distortions and Galaxy Clustering

The standard model of cosmology describes the complex large scale structure of the Universe through less than 10 free parameters. However, concordance with observations requires that about 95\% of the energy content of the universe is invisible to us. Most of this energy is postulated to be in the form of a cosmological constant, $\Lambda$, which drives the observed accelerated expansion of the Universe. Its nature is, however, unknown. This mystery forces cosmologists to look for inconsistencies between theory and data, searching for clues. But finding statistically significant contradictions requires extremely accurate measurements of the composition of the Universe, which are at present limited by our inability to extract all the information contained in the data, rather than being limited by the data itself. In this Thesis, we study how we can overcome these limitations by i) modelling how galaxies cluster on small scales with simulation-based methods, where perturbation theory fails to provide accurate predictions, and ii) developing summary statistics of the density field that are capable of extracting more information than the commonly used two-point functions. In the first half, we show how the real to redshift space mapping can be modelled accurately by going beyond the Gaussian approximation for the pairwise velocity distribution. We then show that simulation-based models can accurately predict the full shape of galaxy clustering in real space, increasing the constraining power on some of the cosmological parameters by a factor of 2 compared to perturbation theory methods. In the second half, we measure the information content of density dependent clustering. We show that it can improve the constraints on all cosmological parameters by factors between 3 and 8 over the two-point function. In particular, exploiting the environment dependence can constrain the mass of neutrinos by a factor of 8$ better than the two-point correlation function alone. We hope that the techniques described in this thesis will contribute to extracting all the cosmological information contained in ongoing and upcoming galaxy surveys, and provide insight into the nature of the accelerated expansion of the universe

Active matter beyond mean-field: Ring-kinetic theory for self-propelled particles

A ring-kinetic theory for Vicsek-style models of self-propelled agents is derived from the exact N-particle evolution equation in phase space. The theory goes beyond mean-field and does not rely on Boltzmann's approximation of molecular chaos. It can handle pre-collisional correlations and cluster formation which both seem important to understand the phase transition to collective motion. We propose a diagrammatic technique to perform a small density expansion of the collision operator and derive the first two equations of the BBGKY-hierarchy. An algorithm is presented that numerically solves the evolution equation for the two-particle correlations on a lattice. Agent-based simulations are performed and informative quantities such as orientational and density correlation functions are compared with those obtained by ring-kinetic theory. Excellent quantitative agreement between simulations and theory is found at not too small noises and mean free paths. This shows that there is parameter ranges in Vicsek-like models where the correlated closure of the BBGKY-hierarchy gives correct and nontrivial results. We calculate the dependence of the orientational correlations on distance in the disordered phase and find that it seems to be consistent with a power law with exponent around -1.8, followed by an exponential decay. General limitations of the kinetic theory and its numerical solution are discussed