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New concept for quantification of similarity relates entropy and energy of objects: First and Second Law entangled, equivalence of temperature and time proposed

Zimak, Petr; Terenzi, Silvia; Strazewski, Peter
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 15/06/2009 Português
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When the difference between changes in energy and entropy at a given temperature is correlated with the ratio between the same changes in energy and entropy at zero average free energy of an ensemble of similar but distinct molecule-sized objects, a highly significant linear dependence results from which a relationship between energy and entropy is derived and the degree of similarity between the distinctly different members within the group of objects can be quantified. This fundamental energy-entropy relationship is likely to be of general interest in physics, most notably in particle physics and cosmology. We predict a consistent and testable way of classifying mini black holes, to be generated in future Large Hadron Collider experiments, by their gravitational energy and area entropy. For any isolated universe we propose absolute temperature and absolute time to be equivalent, much in the same way as energy and entropy are for an isolated ensemble of similar objects. According to this principle, the cosmological constant is the squared product of absolute time and absolute temperature. The symmetry break into a time component and a temperature component of the universe takes place when the first irreversible movement occurs owing to a growing accessed total volume.; Comment: 11 pages text plus 2 pages containing 2 low-res b&w figures (pdf...

Diverging probability density functions for flat-top solitary waves

Peleg, Avner; Chung, Yeojin; Dohnal, Tomáš; Nguyen, Quan M.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 16/06/2009 Português
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We investigate the statistics of flat-top solitary wave parameters in the presence of weak multiplicative dissipative disorder. We consider first propagation of solitary waves of the cubic-quintic nonlinear Schr\"odinger equation (CQNLSE) in the presence of disorder in the cubic nonlinear gain. We show by a perturbative analytic calculation and by Monte Carlo simulations that the probability density function (PDF) of the amplitude $\eta$ exhibits loglognormal divergence near the maximum possible amplitude $\eta_{m}$, a behavior that is similar to the one observed earlier for disorder in the linear gain [A. Peleg et al., Phys. Rev. E {\bf 72}, 027203 (2005)]. We relate the loglognormal divergence of the amplitude PDF to the super-exponential approach of $\eta$ to $\eta_{m}$ in the corresponding deterministic model with linear/nonlinear gain. Furthermore, for solitary waves of the derivative CQNLSE with weak disorder in the linear gain both the amplitude and the group velocity $\beta$ become random. We therefore study analytically and by Monte Carlo simulations the PDF of the parameter $p$, where $p=\eta/(1-\varepsilon_s\beta/2)$ and $\varepsilon_s$ is the self-steepening coefficient. Our analytic calculations and numerical simulations show that the PDF of $p$ is loglognormally divergent near the maximum $p$-value.; Comment: 9 pages...

Exploring the non-linear density field in the Millennium simulations with tessellations - I. The probability distribution function

Pandey, Biswajit; White, Simon D. M.; Springel, Volker; Angulo, Raul
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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We use the Delaunay Tessellation Field Estimator (DTFE) to study the one-point density distribution functions of the Millennium (MS) and Millennium-II (MS-II) simulations. The DTFE technique is based directly on the particle positions, without requiring any type of smoothing or analysis grid, thereby providing high sensitivity to all non-linear structures resolved by the simulations. In order to identify the detailed origin of the shape of the one-point density probability distribution function (PDF), we decompose the simulation particles according to the mass of their host FoF halos, and examine the contributions of different halo mass ranges to the global density PDF. We model the one-point distribution of the FoF halos in each halo mass bin with a set of Monte Carlo realizations of idealized NFW dark matter halos, finding that this reproduces the measurements from the N-body simulations reasonably well, except for a small excess present in simulation results. This excess increases with increasing halo mass. We show that its origin lies in substructure, which becomes progressively more abundant and better resolved in more massive dark matter halos. We demonstrate that the high density tail of the one-point distribution function in less massive halos is severely affected by the gravitational softening length and the mass resolution. In particular...

The intergalactic medium thermal history at redshift z=1.7--3.2 from the Lyman alpha forest: a comparison of measurements using wavelets and the flux distribution

Garzilli, A.; Bolton, J. S.; Kim, T. -S.; Leach, S.; Viel, M.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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We investigate the thermal history of the intergalactic medium (IGM) in the redshift interval z=1.7--3.2 by studying the small-scale fluctuations in the Lyman alpha forest transmitted flux. We apply a wavelet filtering technique to eighteen high resolution quasar spectra obtained with the Ultraviolet and Visual Echelle Spectrograph (UVES), and compare these data to synthetic spectra drawn from a suite of hydrodynamical simulations in which the IGM thermal state and cosmological parameters are varied. From the wavelet analysis we obtain estimates of the IGM thermal state that are in good agreement with other recent, independent wavelet-based measurements. We also perform a reanalysis of the same data set using the Lyman alpha forest flux probability distribution function (PDF), which has previously been used to measure the IGM temperature-density relation. This provides an important consistency test for measurements of the IGM thermal state, as it enables a direct comparison of the constraints obtained using these two different methodologies. We find the constraints obtained from wavelets and the flux PDF are formally consistent with each other, although in agreement with previous studies, the flux PDF constraints favour an isothermal or inverted IGM temperature-density relation. We also perform a joint analysis by combining our wavelet and flux PDF measurements...

PDF4LHC recommendations for LHC Run II

Butterworth, Jon; Carrazza, Stefano; Cooper-Sarkar, Amanda; De Roeck, Albert; Feltesse, Joel; Forte, Stefano; Gao, Jun; Glazov, Sasha; Huston, Joey; Kassabov, Zahari; McNulty, Ronan; Morsch, Andreas; Nadolsky, Pavel; Radescu, Voica; Rojo, Juan; Thorne, Ro
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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We provide an updated recommendation for the usage of sets of parton distribution functions (PDFs) and the assessment of PDF and PDF+$\alpha_s$ uncertainties suitable for applications at the LHC Run II. We review developments since the previous PDF4LHC recommendation, and discuss and compare the new generation of PDFs, which include substantial information from experimental data from the Run I of the LHC. We then propose a new prescription for the combination of a suitable subset of the available PDF sets, which is presented in terms of a single combined PDF set. We finally discuss tools which allow for the delivery of this combined set in terms of optimized sets of Hessian eigenvectors or Monte Carlo replicas, and their usage, and provide some examples of their application to LHC phenomenology.; Comment: 65 pages, 20 figures, additional benchmark comparison plots available from http://www.hep.ucl.ac.uk/pdf4lhc/mc2h-gallery/website. v2, accepted for publication in Journal of Physics G

Gravitational fluctuations of the galaxy distribution

Labini, Francesco Sylos
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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We study the statistical properties of the gravitational field generated by galaxy distribution observed bythe Sloan Digital Sky Survey (DR7). We characterize the probability density function of gravitational force fluctuations and relate its limiting behaviors to the correlation properties of the underlying density field. In addition, we study whether the PDF converges to an asymptotic shape within sample volumes. We consider several volume-limited samples of the Sloan Digital Sky Survey and we compute the gravitational force probability density function (PDF). The gravitational force is computed in spheres of varying radius as is its PDF. We find that (i) the PDF of the force displays features that can be understood in terms of galaxy two-point correlations and (ii) density fluctuations on the largest scales probed, i.e. r~100 Mpc/h, still contribute significantly to the amplitude of the gravitational force. Our main conclusion is that fluctuations in the gravitational force field generated by galaxy structures are also relevant on scales ~ 100 Mpc/h. By assuming that the gravitational fluctuations in the galaxy distribution reflect those in the whole matter distribution, and that peculiar velocities and accelerations are simply correlated...

The role of presumed probability density function in the simulation of non premixed turbulent combustion

Coclite, Alessandro; Pascazio, Giuseppe; De Palma, Pietro; Cutrone, Luigi
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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Flamelet Progress Variable (FPV) combustion models allow the evaluation of all thermo chemical quantities in a reacting flow by computing only the mixture fraction Z and a progress variable C. When using such a method to predict a turbulent combustion in conjunction with a turbulence model, a probability density function (PDF) is required to evaluate statistical averages (e.g., Favre average) of chemical quantities. The choice of the PDF is a compromise between computational costs and accuracy level. The aim of this paper is to investigate the influence of the PDF choice and its modeling aspects in the simulation of non premixed turbulent combustion. Three different models are considered: the standard one, based on the choice of a beta distribution for Z and a Dirac distribution for C; a model employing a beta distribution for both Z and C; a third model obtained using a beta distribution for Z and the statistical most likely distribution (SMLD) for C. The standard model, although widely used, doesn't take into account the interaction between turbulence and chemical kinetics as well as the dependence of the progress variable not only on its mean but also on its variance. The SMLD approach establishes a systematic framework to incorporate informations from an arbitrary number of moments...

Evolution of the cosmological density distribution function

Taylor, Andrew; Watts, Peter
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 07/01/2000 Português
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We present a new calculation for the evolution of the 1-point Probability Distribution Function (PDF) of the cosmological density field based on an exact statistical treatment. Using the Chapman-Kolmogorov equation and second-order Eulerian perturbation theory we propagate the initial density distribution into the nonlinear regime. Our calculations yield the moment generating function, allowing a straightforward derivation of the skewness of the PDF to second order. We find a new dependency on the initial perturbation spectrum. We compare our results with other approximations to the 1-pt PDF, and with N-body simulations. We find that our distribution accurately models the evolution of the 1-pt PDF of dark matter.; Comment: 7 pages, 7 figures, accepted for publication in MNRAS

Oscillatory dynamics of the classical Nonlinear Schrodinger equation

Agafontsev, D. S.; Zakharov, V. E.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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We study numerically the statistical properties of the modulation instability (MI) developing from condensate solution seeded by weak, statistically homogeneous in space noise, in the framework of the classical (integrable) one-dimensional Nonlinear Schrodinger (NLS) equation. We demonstrate that in the nonlinear stage of the MI the moments of the solutions amplitudes oscillate with time around their asymptotic values very similar to sinusoidal law. The amplitudes of these oscillations decay with time $t$ as $t^{-3/2}$, the phases contain the nonlinear phase shift that decays as $t^{-1/2}$, and the period of the oscillations is equal to $\pi$. The asymptotic values of the moments correspond to Rayleigh probability density function (PDF) of waves amplitudes appearance. We show that such behavior of the moments is governed by oscillatory-like, decaying with time, fluctuations of the PDF around the Rayleigh PDF; the time dependence of the PDF turns out to be very similar to that of the moments. We study how the oscillations that we observe depend on the initial noise properties and demonstrate that they should be visible for a very wide variety of statistical distributions of noise.; Comment: 7 pages, 5 figures

Probability density function of turbulent velocity fluctuation

Mouri, H.; Takaoka, M.; Hori, A.; Kawashima, Y.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 11/03/2002 Português
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The probability density function (PDF) of velocity fluctuations is studied experimentally for grid turbulence in a systematical manner. At small distances from the grid, where the turbulence is still developing, the PDF is sub-Gaussian. At intermediate distances, where the turbulence is fully developed, the PDF is Gaussian. At large distances, where the turbulence has decayed, the PDF is hyper-Gaussian. The Fourier transforms of the velocity fluctuations always have Gaussian PDFs. At intermediate distances from the grid, the Fourier transforms are statistically independent of each other. This is the necessary and sufficient condition for Gaussianity of the velocity fluctuations. At small and large distances, the Fourier transforms are dependent.; Comment: 7 pages, 8 figures in a PS file, to appear in Physical Review E

Single or combined location measurements of the same parameter without prior probability. (Altern. title, Parametric inference as useful illusion; Part 1)

Kahrimanis, George
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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Motivation. This version is based solely on the calculus of probability, excluding any statistical principle. "Location measurement" means the pdf of the error is known. When the datum is obtained, intuition suggests something like a pdf for the parameter; here we attempt a critical examination of its meaning. Summary. In default of prior probability the parameter is not defined as a random variable, hence there can be no genuine prior-free parametric inference. Nevertheless prior-free predictive inference regarding any future datum is generated directly from the datum of a location measurement. Such inference turns out as if obtained from a certain pdf ("fiducial") indirectly associated with the parameter. This false pdf can expedite predictive inference, but is inappropriate in the analysis of combined measurements (unless they all are location measurements of the same parameter). Also it has the same distribution as the ostensible Bayesian posterior from a uniform "prior". However, if any of these spurious entities is admitted in the analysis, inconsistent results follow. When we combine measurements, we find that the quantisation errors, inevitable in data recording, must be taken into consideration. These errors cannot be folded into predictive inference in an exact sense; that is...

Parton distributions with LHC data

Ball, Richard D.; Bertone, Valerio; Carrazza, Stefano; Deans, Christopher S.; Del Debbio, Luigi; Forte, Stefano; Guffanti, Alberto; Hartland, Nathan P.; Latorre, Jose I.; Rojo, Juan; Ubiali, Maria
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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We present the first determination of parton distributions of the nucleon at NLO and NNLO based on a global data set which includes LHC data: NNPDF2.3. Our data set includes, besides the deep inelastic, Drell-Yan, gauge boson production and jet data already used in previous global PDF determinations, all the relevant LHC data for which experimental systematic uncertainties are currently available: ATLAS and LHCb W and Z lepton rapidity distributions from the 2010 run, CMS W electron asymmetry data from the 2011 run, and ATLAS inclusive jet cross-sections from the 2010 run. We introduce an improved implementation of the FastKernel method which allows us to fit to this extended data set, and also to adopt a more effective minimization methodology. We present the NNPDF2.3 PDF sets, and compare them to the NNPDF2.1 sets to assess the impact of the LHC data. We find that all the LHC data are broadly consistent with each other and with all the older data sets included in the fit. We present predictions for various standard candle cross-sections, and compare them to those obtained previously using NNPDF2.1, and specifically discuss the impact of ATLAS electroweak data on the determination of the strangeness fraction of the proton. We also present collider PDF sets...

Extracting the Pair Distribution Function of Liquids and Liquid-Vapor Surfaces by Grazing Incidence X-ray Diffraction Mode

Vaknin, David; Bu, Wei; Travesset, Alex
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 22/07/2008 Português
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We show that the structure factor $S(q)$ of water can be obtained from X-ray synchrotron experiments at grazing angle of incidence (in reflection mode) by using a liquid surface diffractometer. The corrections used to obtain $S(q)$ self-consistently are described. Applying these corrections to scans at different incident beam angles (above the critical angle) collapses the measured intensities into a single master curve, without fitting parameters, which within a scale factor yields $S(q)$. Performing the measurements below the critical angle for total reflectivity yields the structure factor of the top most layers of the water/vapor interface. Our results indicate water restructuring at the vapor/water interface. We also introduce a new approach to extract $g(r)$, the pair distribution function (PDF), by expressing the PDF as a linear sum of Error functions whose parameters are refined by applying a non-linear least square fit method. This approach enables a straightforward determination of the inherent uncertainties in the PDF. Implications of our results to previously measured and theoretical predictions of the PDF are also discussed.

Shock generated vorticity in the interstellar medium and the origin of the stellar initial mass function

Kevlahan, N.; Pudritz, Ralph E.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 16/07/2009 Português
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The main observational evidence for turbulence in the interstellar medium (ISM) and molecular clouds is the power-law energy spectrum for velocity fluctuations, E(k) \propto k^{\alpha}. The Kolmogorov scaling exponent, \alpha=-5/3, is typical. At the same time, the observed probability distribution function (PDF) of gas densities in both the ISM as well as in molecular clouds is a log-normal distribution, which is similar to the initial mass function (IMF) that describes the distribution of stellar masses. We examine the density and velocity structure of interstellar gas traversed by curved shock waves in the kinematic limit. We demonstrate mathematically that just a few passages of curved shock waves generically produces a log-normal density PDF. This explains the ubiquity of the log-normal PDF in many different numerical simulations. We also show that subsequent interaction with a spherical blast wave generates a power-law density distribution at high densities, qualitatively similar to the Salpeter power-law for the IMF. Finally, we show that a focused shock produces a downstream flow with energy spectrum exponent \alpha=-2. Subsequent shock passages reduce this slope, achieving \alpha \approx -5/3 after a few passages. Subsequent dissipation of energy piled up at the small scales will act to maintain the spectrum very near to the Kolomogorov value. Therefore...

A positivity-preserving high-order semi-Lagrangian discontinuous Galerkin scheme for the Vlasov-Poisson equations

Rossmanith, J. A.; Seal, D. C.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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The Vlasov-Poisson equations describe the evolution of a collisionless plasma, represented through a probability density function (PDF) that self-interacts via an electrostatic force. One of the main difficulties in numerically solving this system is the severe time-step restriction that arises from parts of the PDF associated with moderate-to-large velocities. The dominant approach in the plasma physics community for removing these time-step restrictions is the so-called particle-in-cell (PIC) method, which discretizes the distribution function into a set of macro-particles, while the electric field is represented on a mesh. Several alternatives to this approach exist, including fully Lagrangian, fully Eulerian, and so-called semi-Lagrangian methods. The focus of this work is the semi-Lagrangian approach, which begins with a grid-based Eulerian representation of both the PDF and the electric field, then evolves the PDF via Lagrangian dynamics, and finally projects this evolved field back onto the original Eulerian mesh. In particular, we develop in this work a method that discretizes the 1+1 Vlasov-Poisson system via a high-order discontinuous Galerkin (DG) method in phase space, and an operator split, semi-Lagrangian method in time. Second-order accuracy in time is relatively easy to achieve via Strang operator splitting. With additional work...

Craig's XY-distribution and the statistics of Lagrangian power in two-dimensional turbulence

Bandi, M.; Connaughton, C.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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We examine the probability distribution function (pdf) of energy injection rate (power) in numerical simulations of stationary two--dimensional (2D) turbulence in the Lagrangian frame. The simulation is designed to mimic an electromagnetically driven fluid layer, a well-documented system for generating two--dimensional turbulence in the laboratory. In our simulations, the forcing and velocity fields are close to Gaussian. On the other hand, the measured PDF of injected power is very sharply peaked at zero, suggestive of a singularity there, with tails which are exponential but asymmetric. Large positive fluctuations are more probable than large negative fluctuations. It is this asymmetry of the tails, which leads to a net positive mean value for the energy input despite the most probable value being zero. The main features of the power distribution are well described by Craig's XY distribution for the PDF of the product of two correlated normal variables. We show that the power distribution should exhibit a logarithmic singularity at zero and decay exponentially for large absolute values of the power. We calculate the asymptotic behavior and express the asymmetry of the tails in terms of the correlation coefficient of the force and velocity. We compare the measured pdfs with the theoretical calculations and briefly discuss how the power pdf might change with other forcing mechanisms.; Comment: 9 pages...

Evaluate the Word Error Rate of Binary Block Codes with Square Radius Probability Density Function

Chen, Xiaogang; Yang, Hongwen; Gu, Jian; Yang, Hongkui
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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The word error rate (WER) of soft-decision-decoded binary block codes rarely has closed-form. Bounding techniques are widely used to evaluate the performance of maximum-likelihood decoding algorithm. But the existing bounds are not tight enough especially for low signal-to-noise ratios and become looser when a suboptimum decoding algorithm is used. This paper proposes a new concept named square radius probability density function (SR-PDF) of decision region to evaluate the WER. Based on the SR-PDF, The WER of binary block codes can be calculated precisely for ML and suboptimum decoders. Furthermore, for a long binary block code, SR-PDF can be approximated by Gamma distribution with only two parameters that can be measured easily. Using this property, two closed-form approximative expressions are proposed which are very close to the simulation results of the WER of interesting.; Comment: I tried to fix the problem that figures cannot be displayed in Version1~3, but it doesnot work. Some new results about decoding threshold is added in version 4 and a pdf is included which is compile in my PC. Figures can displayed

Density probability distribution in one-dimensional polytropic gas dynamics

Passot, Thierry; Vazquez-Semadeni, Enrique
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 10/02/1998 Português
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We discuss the generation and statistics of the density fluctuations in highly compressible polytropic turbulence, based on a simple model and one-dimensional numerical simulations. Observing that density structures tend to form in a hierarchical manner, we assume that density fluctuations follow a random multiplicative process. When the polytropic exponent $\gamma$ is equal to unity, the local Mach number is independent of the density, and our assumption leads us to expect that the probability density function (PDF) of the density field is a lognormal. This isothermal case is found to be singular, with a dispersion $\sigma_s^2$ which scales like the square turbulent Mach number $\tilde M^2$, where $s\equiv \ln \rho$ and $\rho$ is the fluid density. This leads to much higher fluctuations than those due to shock jump relations. Extrapolating the model to the case $\gamma \not =1$, we find that, as the Mach number becomes large, the density PDF is expected to asymptotically approach a power-law regime, at high densities when $\gamma<1$, and at low densities when $\gamma>1$. This effect can be traced back to the fact that the pressure term in the momentum equation varies exponentially with $s$, thus opposing the growth of fluctuations on one side of the PDF...

The PDF4LHC Working Group Interim Recommendations

Botje, Michiel; Butterworth, Jon; Cooper-Sarkar, Amanda; de Roeck, Albert; Feltesse, Joel; Forte, Stefano; Glazov, Alexander; Huston, Joey; McNulty, Ronan; Sjostrand, Torbjorn; Thorne, Robert
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 03/01/2011 Português
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This note provides an interim summary of the current recommendations of the PDF4LHC working group for the use of parton distribution functions (PDFs) and of PDF uncertainties at the LHC, for cross section and cross section uncertainty calculations. It also contains a succinct user guide to the computation of PDF uncertainties and correlations using available PDF sets. A companion note (the PDF4LHC Working Group Interim Report) summarizes predictions for benchmark cross sections at the LHC (7 TeV) at NLO using modern PDFs currently available from 6 PDF fitting groups.; Comment: 12 pages

The Probability Distribution Function of Column Density in Molecular Clouds

Vazquez-Semadeni, Enrique; Garcia, Nieves
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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(Abridged) We discuss the probability distribution function (PDF) of column density resulting from density fields with lognormal PDFs, applicable to isothermal gas (e.g., probably molecular clouds). We suggest that a ``decorrelation length'' can be defined as the distance over which the density auto-correlation function has decayed to, for example, 10% of its zero-lag value, so that the density ``events'' along a line of sight can be assumed to be independent over distances larger than this, and the Central Limit Theorem should be applicable. However, using random realizations of lognormal fields, we show that the convergence to a Gaussian is extremely slow in the high- density tail. Thus, the column density PDF is not expected to exhibit a unique functional shape, but to transit instead from a lognormal to a Gaussian form as the ratio $\eta$ of the column length to the decorrelation length increases. Simultaneously, the PDF's variance decreases. For intermediate values of $\eta$, the column density PDF assumes a nearly exponential decay. We then discuss the density power spectrum and the expected value of $\eta$ in actual molecular clouds. Observationally, our results suggest that $\eta$ may be inferred from the shape and width of the column density PDF in optically-thin-line or extinction studies. Our results should also hold for gas with finite-extent power-law underlying density PDFs...