12 research outputs found
Finite Number and Finite Size Effects in Relativistic Bose-Einstein Condensation
Bose-Einstein condensation of a relativistic ideal Bose gas in a rectangular
cavity is studied. Finite size corrections to the critical temperature are
obtained by the heat kernel method. Using zeta-function regularization of
one-loop effective potential, lower dimensional critical temperatures are
calculated. In the presence of strong anisotropy, the condensation is shown to
occur in multisteps. The criteria of this behavior is that critical
temperatures corresponding to lower dimensional systems are smaller than the
three dimensional critical temperature.Comment: 18 pages, 9 figures, Fig.3 replaced, to appear in Physical Review
Random Convex Hulls and Extreme Value Statistics
In this paper we study the statistical properties of convex hulls of
random points in a plane chosen according to a given distribution. The points
may be chosen independently or they may be correlated. After a non-exhaustive
survey of the somewhat sporadic literature and diverse methods used in the
random convex hull problem, we present a unifying approach, based on the notion
of support function of a closed curve and the associated Cauchy's formulae,
that allows us to compute exactly the mean perimeter and the mean area enclosed
by the convex polygon both in case of independent as well as correlated points.
Our method demonstrates a beautiful link between the random convex hull problem
and the subject of extreme value statistics. As an example of correlated
points, we study here in detail the case when the points represent the vertices
of independent random walks. In the continuum time limit this reduces to
independent planar Brownian trajectories for which we compute exactly, for
all , the mean perimeter and the mean area of their global convex hull. Our
results have relevant applications in ecology in estimating the home range of a
herd of animals. Some of these results were announced recently in a short
communication [Phys. Rev. Lett. {\bf 103}, 140602 (2009)].Comment: 61 pages (pedagogical review); invited contribution to the special
issue of J. Stat. Phys. celebrating the 50 years of Yeshiba/Rutgers meeting