Two-dimensional active motion

The diffusion in two dimensions of non-interacting active particles that follow an arbitrary motility pattern is considered for analysis. Accordingly, the transport equation is generalized to take into account an arbitrary distribution of scattered angles of the swimming direction, which encompasses the pattern of motion of particles that move at constant speed. An exact analytical expression for the marginal probability density of finding a particle on a given position at a given instant, independently of its direction of motion, is provided; and a connection with a generalized diffusion equation is unveiled. Exact analytical expressions for the time dependence of the mean-square displacement and of the kurtosis of the distribution of the particle positions are presented. For this, it is shown that only the first trigonometric moments of the distribution of the scattered direction of motion are needed. The effects of persistence and of circular motion are discussed for different families of distributions of the scattered direction of motion.Comment: 17 pages, 8 figures (published version

Probability distributions for Poisson processes with pile-up

In this paper, two parametric probability distributions capable to describe the statistics of X-ray photon detection by a CCD are presented. They are formulated from simple models that account for the pile-up phenomenon, in which two or more photons are counted as one. These models are based on the Poisson process, but they have an extra parameter which includes all the detailed mechanisms of the pile-up process that must be fitted to the data statistics simultaneously with the rate parameter. The new probability distributions, one for number of counts per time bins (Poisson-like), and the other for waiting times (exponential-like) are tested fitting them to statistics of real data, and between them through numerical simulations, and their results are analyzed and compared. The probability distributions presented here can be used as background statistical models to derive likelihood functions for statistical methods in signal analysis

Thermodynamics of the Relativistic Fermi gas in D Dimensions

The influence of spatial dimensionality and particle-antiparticle pair production on the thermodynamic properties of the relativistic Fermi gas, at finite chemical potential, is studied. Resembling a kind of phase transition, qualitatively different behaviors of the thermodynamic susceptibilities, namely the isothermal compressibility and the specific heat, are markedly observed at different temperature regimes as function of the system dimensionality and of the rest mass of the particles. A minimum in the isothermal compressibility marks a characteristic temperature, in the range of tenths of the Fermi temperature, at which the system transit from a normal phase, to a phase where the gas compressibility grows as a power law of the temperature. Curiously, we find that for a particle density of a few times the density of nuclear matter, and rest masses of the order of 10 MeV, the minimum of the compressibility occurs at approximately 170 MeV/k, which roughly estimates the critical temperature of hot fermions as those occurring in the gluon-quark plasma phase transition.Comment: 23 pages, 5 figures, Submitted for publicatio

Emergence of collective motion in a model of interacting Brownian particles

By studying a system of Brownian particles, interacting only through a local social-like force (velocity alignment), we show that self-propulsion is not a necessary feature for the flocking transition to take place as long as underdamped particle dynamics can be guaranteed. Moreover, the system transits from stationary phases close to thermal equilibrium, with no net flux of particles, to far-from-equilibrium ones exhibiting collective motion, long-range order and giant number fluctuations, features typically associated to ordered phases of models where self-propulsion is considered.Comment: 5 pages, 2 figure

Smoluchowski Diffusion Equation for Active Brownian Swimmers

We study the free diffusion in two dimensions of active-Brownian swimmers subject to passive fluctuations on the translational motion and to active fluctuations on the rotational one. The Smoluchowski equation is derived from a Langevin-like model of active swimmers, and analytically solved in the long-time regime for arbitrary values of the P\'eclet number, this allows us to analyze the out-of-equilibrium evolution of the positions distribution of active particles at all time regimes. Explicit expressions for the mean-square displacement and for the kurtosis of the probability distribution function are presented, and the effects of persistence discussed. We show through Brownian dynamics simulations that our prescription for the mean-square displacement gives the exact time dependence at all times. The departure of the probability distribution from a Gaussian, measured by the kurtosis, is also analyzed both analytically and computationally. We find that for P\'eclet numbers $\lesssim 0.1$, the distance from Gaussian increases as $\sim t^{-2}$ at short times, while it diminishes as $\sim t^{-1}$ in the asymptotic limit.Comment: The misspelled name of an author has been correcte

Active motion on curved surfaces

A theoretical analysis of active motion on curved surfaces is presented in terms of a generalization of the Telegrapher's equation. Such generalized equation is explicitly derived as the polar approximation of the hierarchy of equations obtained from the corresponding Fokker-Planck equation of active particles diffusing on curved surfaces. The general solution to the generalized telegrapher's equation is given for a pulse with vanishing current as initial data. Expressions for the probability density and the mean squared geodesic-displacement are given in the limit of weak curvature. As an explicit example of the formulated theory, the case of active motion on the sphere is presented, where oscillations observed in the mean squared geodesic-displacement are explained.Comment: Manuscript submitted, 12 pages, two figure

Revisiting the concept of chemical potential in classical and quantum gases: A perspective from Equilibrium Statistical Mechanics

In this work we revisit the concept of chemical potential $\mu$ in both classical and quantum gases from a perspective of Equilibrium Statistical Mechanics (ESM). Two new results regarding the equation of state $\mu=\mu(n,T)$, where $n$ is the particle density and $T$ the absolute temperature, are given for the classical interacting gas and for the weakly-interacting quantum Bose gas. In order to make this review self-contained and adequate for a general reader we provide all the basic elements in an advanced-undergraduate or graduate statistical mechanics course required to follow all the calculations. We start by presenting a calculation of $\mu(n,T)$ for the classical ideal gas in the canonical ensemble. After this, we consider the interactions between particles and compute the effects of them on $\mu(n,T)$ for the van der Waals gas. For quantum gases we present an alternative approach to calculate the Bose-Einstein (BE) and Fermi-Dirac (FD) statistics. We show that this scheme can be straightforwardly generalized to determine what we have called Intermediate Quantum Statistics (IQS) which deal with ideal quantum systems where a single-particle energy can be occupied by at most $j$ particles with $0 \leqslant j \leqslant N$ with $N$ the total number of particles. In the final part we address general considerations that underlie the theory of weakly interacting quantum gases. In the case of the weakly interacting Bose gas, we focus our attention to the equation of state $\mu=\mu(n,T)$ in the Hartree-Fock mean-field approximation (HF) and the implications of such results in the elucidation of the order of the phase transitions involved in the BEC phase for non-ideal Bose gases.Comment: 43 pages, 5 figures. The following article has been submitted to the American Journal of Physics. After it is published, it will be found at http://scitation.aip.org/ajp

Hyperelliptic curves of genus 3 with prescribed automorphism group

We study genus 3 hyperelliptic curves which have an extra involution. The locus \L_3 of these curves is a 3-dimensional subvariety in the genus 3 hyperelliptic moduli \H_3. We find a birational parametrization of this locus by affine 3-space. For every moduli point \p \in \H_3 such that |\Aut (\p)|>2, the field of moduli is a field of definition. We provide a rational model of the curve over its field of moduli for all moduli points \p \in \H_3 such that |\Aut(\p)|>4. This is the first time that such a rational model of these curves appears in the literature

Covering Rational Ruled Surfaces

We present an algorithm that covers any given rational ruled surface with two rational parametrizations. In addition, we present an algorithm that transforms any rational surface parametrization into a new rational surface parametrization without affine base points and such that the degree of the corresponding maps is preserved.Comment: 19 pages, 2 figures in jpg. v2: minor correction of Example 1. v3: updated acknowledgement

Active Particles Moving in Two-Dimensional Space with Constant Speed: Revisiting the Telegrapher's Equation

Starting from a Langevin description of active particles that move with constant speed in infinite two-dimensional space and its corresponding Fokker-Planck equation, we develop a systematic method that allows us to obtain the coarse-grained probability density of finding a particle at a given location and at a given time to arbitrary short time regimes. By going beyond the diffusive limit, we derive a novel generalization of the telegrapher's equation. Such generalization preserves the hyperbolic structure of the equation and incorporates memory effects on the diffusive term. While no difference is observed for the mean square displacement computed from the two-dimensional telegrapher's equation and from our generalization, the kurtosis results into a sensible parameter that discriminates between both approximations. We carried out a comparative analysis in Fourier space that shed light on why the telegrapher's equation is not an appropriate model to describe the propagation of particles with constant speed in dispersive media.Comment: 19 pages, 3 figure