7 research outputs found
Mean Field Theory of Spherical Gravitating Systems
Important gaps remain in our understanding of the thermodynamics and
statistical physics of self-gravitating systems. Using mean field theory, here
we investigate the equilibrium properties of several spherically symmetric
model systems confined in a finite domain consisting of either point masses, or
rotating mass shells of different dimension. We establish a direct connection
between the spherically symmetric equilibrium states of a self-gravitating
point mass system and a shell model of dimension 3. We construct the
equilibrium density functions by maximizing the entropy subject to the usual
constraints of normalization and energy, but we also take into account the
constraint on the sum of the squares of the individual angular momenta, which
is also an integral of motion for these symmetric systems. Two new statistical
ensembles are introduced which incorporate the additional constraint. They are
used to investigate the possible occurrence of a phase transition as the
defining parameters for each ensemble are altered
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.
Metastability, negative specific heat and weak mixing in classical long-range many-rotator system
We perform a molecular dynamical study of the isolated classical
Hamiltonian , known to
exhibit a second order phase transition, being disordered for and ordered otherwise ( total energy
and ). We focus
on the nonextensive case and observe that, for , a
basin of attraction exists for the initial conditions for which the system
quickly relaxes onto a longstanding metastable state (whose duration presumably
diverges with like ) which eventually crosses over to the
microcanonical Boltzmann-Gibbs stable state. The temperature associated with
the (scaled) average kinetic energy per particle is lower in the metastable
state than in the stable one. It is exhibited for the first time that the
appropriately scaled maximal Lyapunov exponent
, where, for all values of ,
numerically coincides with {\it one third} of its value for , hence
decreases from 1/9 to zero when increases from zero to unity,
remaining zero thereafter. This new and simple {\it connection between
anomalies above and below the critical point} reinforces the nonextensive
universality scenario.Comment: 9 pages and 4 PS figure