A numerical comparison of theories of violent relaxation

Using N-body simulations with a large set of massless test particles we compare the predictions of two theories of violent relaxation, the well known Lynden-Bell theory and the more recent theory by Nakamura. We derive ``weaken'' versions of both theories in which we use the whole equilibrium coarse-grained distribution function as a constraint instead of the total energy constraint. We use these weaken theories to construct expressions for the conditional probability $K_i(\tau)$ that a test particle initially at the phase-space coordinate $\tau$ would end-up in the $i$'th macro-cell at equilibrium. We show that the logarithm of the ratio $R_{ij}(\tau) \equiv K_i(\tau)/K_j(\tau)$ is directly proportional to the initial phase-space density $f_0(\tau)$ for the Lynden-Bell theory and inversely proportional to $f_0(\tau)$ for the Nakamura theory. We then measure $R_{ij}(\tau)$ using a set of N-body simulations of a system undergoing a gravitational collapse to check the validity of the two theories of violent relaxation. We find that both theories are at odds with the numerical results, qualitatively and quantitatively.Comment: Replaced with a revised version, which is now accepted to MNRAS. LaTeX, 12 pages, 6 figure

Major Mergers of Galaxy Haloes: Cuspy or Cored Inner Density Profile?

We present the results from a series of collisionless N-body simulations of major mergers of galaxy dark matter haloes with density profiles having either inner cusps or cores. Our simulations range from 2x10^5 to 10^7 particles, allowing us to probe the phase-space distribution of dark matter particles in the innermost regions (less than 0.005 virial radii) of cold dark matter haloes, a subject of much recent debate. We find that a major merger of two cored haloes yields a cored halo and does not result in a cuspy profile seen in many cosmological simulations. This result is unchanged if we consider mergers with parent mass ratios of 3:1 instead of 1:1. Mergers of a cuspy halo with either a cored halo or a second cuspy halo of equal mass, on the other hand, produce cuspy haloes with a slightly reduced inner logarithmic slope. Cuspy haloes, once formed, therefore appear resilient to major mergers. We find the velocity structure of the remnants to be mildly anisotropic, with a Maxwellian velocity distribution near the centre but not in the outer portions of the final haloes. Violent relaxation is effective only during the early phase of mergers, with phase mixing likely to be the dominant relaxation process at late times.Comment: MNRAS (in press). 15 pages, 16 figures; minor changes, one additional figure, updated reference

Fluoro-deoxi-glucose uptake and angiogenesis are independent biological features in lung metastases

Neoangiogenesis and enhanced glucose metabolism in neoplasms are likely to be activated by the same biochemical stimulus; hypoxia. A correlation between these two parameters has been postulated. The objective of this study was to evaluate the relationship between Fluoro-desoxi-glucose uptake at positron emission tomography scan and angiogenesis in lung metastasis. Fluoro-desoxi-glucose activity, expressed as a standard uptake value, and microvessel intratumoural density, were retrospectively calculated in a series of 43 lung metastasis resected in 19 patients. Primary sites were colorectal cancer in 16 metastases, sarcoma in eight, gynaecological in four and other sites in 15. The correlation between the two parameters was tested by logistic regression and multivariate analysis. Positron emission tomography scan was positive in 17 patients (sensitivity 89%). No correlation was observed between standard uptake value and microvessel intratumoural density in this series of lung metastasis. Positron emission tomography negative and positive nodules presented comparable value of microvessel intratumoural density (12.9 vs 11.3). Standard uptake value was significantly correlated with nodules size and was higher in colon cancer metastasis than in sarcoma ones. Microvessel intratumoural density was independent from nodule size but significantly higher in sarcoma than in colon cancer metastasis. The lack of correlation was confirmed by multivariate analysis after adjustment for tumour type and nodules size. The present study demonstrated that positron emission tomography scan is positive in a high proportion of patients regardless of microvessel density. Glucose uptake and angiogenesis appear to be independent biological features in lung metastasis. This observation may have implications for future antiangiogenic therapies