10,757 research outputs found
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 , the distance from Gaussian increases as at short times,
while it diminishes as in the asymptotic limit.Comment: The misspelled name of an author has been correcte
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 in both
classical and quantum gases from a perspective of Equilibrium Statistical
Mechanics (ESM). Two new results regarding the equation of state
, where is the particle density and 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 for the classical ideal gas in the canonical ensemble. After
this, we consider the interactions between particles and compute the effects of
them on 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 particles with with 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
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
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
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
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