Statistical Mechanics of Relativistic One-Dimensional Self-Gravitating Systems

We consider the statistical mechanics of a general relativistic one-dimensional self-gravitating system. The system consists of $N$-particles coupled to lineal gravity and can be considered as a model of $N$ relativistically interacting sheets of uniform mass. The partition function and one-particle distitrubion functions are computed to leading order in $1/c$ where $c$ is the speed of light; as $c\to\infty$ results for the non-relativistic one-dimensional self-gravitating system are recovered. We find that relativistic effects generally cause both position and momentum distribution functions to become more sharply peaked, and that the temperature of a relativistic gas is smaller than its non-relativistic counterpart at the same fixed energy. We consider the large-N limit of our results and compare this to the non-relativistic case.Comment: latex, 60 pages, 22 figure

Dynamical N-body Equlibrium in Circular Dilaton Gravity

We obtain a new exact equilibrium solution to the N-body problem in a one-dimensional relativistic self-gravitating system. It corresponds to an expanding/contracting spacetime of a circle with N bodies at equal proper separations from one another around the circle. Our methods are straightforwardly generalizable to other dilatonic theories of gravity, and provide a new class of solutions to further the study of (relativistic) one-dimensional self-gravitating systems.Comment: 4 pages, latex, reference added, minor changes in wordin

Exact Relativistic Two-Body Motion in Lineal Gravity

We consider the N-body problem in (1+1) dimensional lineal gravity. For 2 point masses (N=2) we obtain an exact solution for the relativistic motion. In the equal mass case we obtain an explicit expression for their proper separation as a function of their mutual proper time. Our solution gives the exact Hamiltonian to infinite order in the gravitational coupling constant.Comment: latex, 11 pages, 2 figures, final version to appear in Phys. Rev. Let

Chaos in an Exact Relativistic 3-body Self-Gravitating System

We consider the problem of three body motion for a relativistic one-dimensional self-gravitating system. After describing the canonical decomposition of the action, we find an exact expression for the 3-body Hamiltonian, implicitly determined in terms of the four coordinate and momentum degrees of freedom in the system. Non-relativistically these degrees of freedom can be rewritten in terms of a single particle moving in a two-dimensional hexagonal well. We find the exact relativistic generalization of this potential, along with its post-Newtonian approximation. We then specialize to the equal mass case and numerically solve the equations of motion that follow from the Hamiltonian. Working in hexagonal-well coordinates, we obtaining orbits in both the hexagonal and 3-body representations of the system, and plot the Poincare sections as a function of the relativistic energy parameter $\eta$. We find two broad categories of periodic and quasi-periodic motions that we refer to as the annulus and pretzel patterns, as well as a set of chaotic motions that appear in the region of phase-space between these two types. Despite the high degree of non-linearity in the relativistic system, we find that the the global structure of its phase space remains qualitatively the same as its non-relativisitic counterpart for all values of $\eta$ that we could study. However the relativistic system has a weaker symmetry and so its Poincare section develops an asymmetric distortion that increases with increasing $\eta$. For the post-Newtonian system we find that it experiences a KAM breakdown for $\eta \simeq 0.26$: above which the near integrable regions degenerate into chaos.Comment: latex, 65 pages, 36 figures, high-resolution figures available upon reques

Determining the date of diagnosis – is it a simple matter? The impact of different approaches to dating diagnosis on estimates of delayed care for ovarian cancer in UK primary care

Background Studies of cancer incidence and early management will increasingly draw on routine electronic patient records. However, data may be incomplete or inaccurate. We developed a generalisable strategy for investigating presenting symptoms and delays in diagnosis using ovarian cancer as an example. Methods The General Practice Research Database was used to investigate the time between first report of symptom and diagnosis of 344 women diagnosed with ovarian cancer between 01/06/2002 and 31/05/2008. Effects of possible inaccuracies in dating of diagnosis on the frequencies and timing of the most commonly reported symptoms were investigated using four increasingly inclusive definitions of first diagnosis/suspicion: 1. "Definite diagnosis" 2. "Ambiguous diagnosis" 3. "First treatment or complication suggesting pre-existing diagnosis", 4 "First relevant test or referral". Results The most commonly coded symptoms before a definite diagnosis of ovarian cancer, were abdominal pain (41%), urogenital problems(25%), abdominal distension (24%), constipation/change in bowel habits (23%) with 70% of cases reporting at least one of these. The median time between first reporting each of these symptoms and diagnosis was 13, 21, 9.5 and 8.5 weeks respectively. 19% had a code for definitions 2 or 3 prior to definite diagnosis and 73% a code for 4. However, the proportion with symptoms and the delays were similar for all four definitions except 4, where the median delay was 8, 8, 3, 10 and 0 weeks respectively. Conclusion Symptoms recorded in the General Practice Research Database are similar to those reported in the literature, although their frequency is lower than in studies based on self-report. Generalisable strategies for exploring the impact of recording practice on date of diagnosis in electronic patient records are recommended, and studies which date diagnoses in GP records need to present sensitivity analyses based on investigation, referral and diagnosis data. Free text information may be essential in obtaining accurate estimates of incidence, and for accurate dating of diagnoses

A dynamical classification of the range of pair interactions

We formalize a classification of pair interactions based on the convergence properties of the {\it forces} acting on particles as a function of system size. We do so by considering the behavior of the probability distribution function (PDF) P(F) of the force field F in a particle distribution in the limit that the size of the system is taken to infinity at constant particle density, i.e., in the "usual" thermodynamic limit. For a pair interaction potential V(r) with V(r) \rightarrow \infty) \sim 1/r^a defining a {\it bounded} pair force, we show that P(F) converges continuously to a well-defined and rapidly decreasing PDF if and only if the {\it pair force} is absolutely integrable, i.e., for a > d-1, where d is the spatial dimension. We refer to this case as {\it dynamically short-range}, because the dominant contribution to the force on a typical particle in this limit arises from particles in a finite neighborhood around it. For the {\it dynamically long-range} case, i.e., a \leq d-1, on the other hand, the dominant contribution to the force comes from the mean field due to the bulk, which becomes undefined in this limit. We discuss also how, for a \leq d-1 (and notably, for the case of gravity, a=d-2) P(F) may, in some cases, be defined in a weaker sense. This involves a regularization of the force summation which is generalization of the procedure employed to define gravitational forces in an infinite static homogeneous universe. We explain that the relevant classification in this context is, however, that which divides pair forces with a > d-2 (or a < d-2), for which the PDF of the {\it difference in forces} is defined (or not defined) in the infinite system limit, without any regularization. In the former case dynamics can, as for the (marginal) case of gravity, be defined consistently in an infinite uniform system.Comment: 12 pages, 1 figure; significantly shortened and focussed, additional references, version to appear in J. Stat. Phy

Effect of angular momentum on equilibrium properties of a self-gravitating system

The microcanonical properties of a two dimensional system of N classical particles interacting via a smoothed Newtonian potential as a function of the total energy E and the total angular momentum L are discussed. In order to estimate suitable observables a numerical method based on an importance sampling algorithm is presented. The entropy surface shows a negative specific heat region at fixed L for all L. Observables probing the average mass distribution are used to understand the link between thermostatistical properties and the spatial distribution of particles. In order to define a phase in non-extensive system we introduce a more general observable than the one proposed by Gross and Votyakov [Eur. Phys. J. B:15, 115 (2000)]: the sign of the largest eigenvalue of the entropy surface curvature. At large E the gravitational system is in a homogeneous gas phase. At low E there are several collapse phases; at L=0 there is a single cluster phase and for L>0 there are several phases with 2 clusters. All these pure phases are separated by first order phase transition regions. The signal of critical behaviour emerges at different points of the parameter space (E,L). We also discuss the ensemble introduced in a recent pre-print by Klinko & Miller; this ensemble is the canonical analogue of the one at constant energy and constant angular momentum. We show that a huge loss of informations appears if we treat the system as a function of intensive parameters: besides the known non-equivalence at first order phase transitions, there exit in the microcanonical ensemble some values of the temperature and the angular velocity for which the corresponding canonical ensemble does not exist, i.e. the partition sum diverges.Comment: 17 pages, 11 figures, submitted to Phys. Rev.

Detection of air trapping in chronic obstructive pulmonary disease by low frequency ultrasound

<p>Abstract</p> <p>Background</p> <p>Spirometry is regarded as the gold standard for the diagnosis of COPD, yet the condition is widely underdiagnosed. Therefore, additional screening methods that are easy to perform and to interpret are needed. Recently, we demonstrated that low frequency ultrasound (LFU) may be helpful for monitoring lung diseases. The objective of this study was to evaluate whether LFU can be used to detect air trapping in COPD. In addition, we evaluated the ability of LFU to detect the effects of short-acting bronchodilator medication.</p> <p>Methods</p> <p>Seventeen patients with COPD and 9 healthy subjects were examined by body plethysmography and LFU. Ultrasound frequencies ranging from 1 to 40 kHz were transmitted to the sternum and received at the back during inspiration and expiration. The high pass frequency was determined from the inspiratory and the expiratory signals and their difference termed ΔF. Measurements were repeated after inhalation of salbutamol.</p> <p>Results</p> <p>We found significant differences in ΔF between COPD subjects and healthy subjects. These differences were already significant at GOLD stage 1 and increased with the severity of COPD. Sensitivity for detection of GOLD stage 1 was 83% and for GOLD stages worse than 1 it was 91%. Bronchodilator effects could not be detected reliably.</p> <p>Conclusions</p> <p>We conclude that low frequency ultrasound is cost-effective, easy to perform and suitable for detecting air trapping. It might be useful in screening for COPD.</p> <p>Trial Registration</p> <p>ClinicalTrials.gov: <a href="http://www.clinicaltrials.gov/ct2/show/NCT01080924">NCT01080924</a></p

Statistics of the gravitational force in various dimensions of space: from Gaussian to Levy laws

We discuss the distribution of the gravitational force created by a Poissonian distribution of field sources (stars, galaxies,...) in different dimensions of space d. In d=3, it is given by a Levy law called the Holtsmark distribution. It presents an algebraic tail for large fluctuations due to the contribution of the nearest neighbor. In d=2, it is given by a marginal Gaussian distribution intermediate between Gaussian and Levy laws. In d=1, it is exactly given by the Bernouilli distribution (for any particle number N) which becomes Gaussian for N>>1. Therefore, the dimension d=2 is critical regarding the statistics of the gravitational force. We generalize these results for inhomogeneous systems with arbitrary power-law density profile and arbitrary power-law force in a d-dimensional universe