23 research outputs found
Saddle-point entropy states of equilibrated self-gravitating systems
In this Letter, we investigate the stability of the statistical equilibrium
of spherically symmetric collisionless self-gravitating systems. By calculating
the second variation of the entropy, we find that perturbations of the relevant
physical quantities should be classified as long- and short-range
perturbations, which correspond to the long- and short-range relaxation
mechanisms, respectively. We show that the statistical equilibrium states of
self-gravitating systems are neither maximum nor minimum, but complex
saddle-point entropy states, and hence differ greatly from the case of ideal
gas. Violent relaxation should be divided into two phases. The first phase is
the entropy-production phase, while the second phase is the entropy-decreasing
phase. We speculate that the second-phase violent relaxation may just be the
long-wave Landau damping, which would work together with short-range
relaxations to keep the system equilibrated around the saddle-point entropy
states.Comment: 5 pages, 1 figure, MNRAS Letter, in the pres
Chaos and Elliptical Galaxies
Recent results on chaos in triaxial galaxy models are reviewed. Central mass
concentrations like those observed in early-type galaxies -- either stellar
cusps, or massive black holes -- render most of the box orbits in a triaxial
potential stochastic. Typical Liapunov times are 3-5 crossing times, and
ensembles of stochastic orbits undergo mixing on time scales that are roughly
an order of magnitude longer. The replacement of the regular orbits by
stochastic orbits reduces the freedom to construct self-consistent equilibria,
and strong triaxiality can be ruled out for galaxies with sufficiently high
central mass concentrations.Comment: uuencoded gziped PostScript, 12 pages including figure
Classical and semi-classical energy conditions
The standard energy conditions of classical general relativity are (mostly)
linear in the stress-energy tensor, and have clear physical interpretations in
terms of geodesic focussing, but suffer the significant drawback that they are
often violated by semi-classical quantum effects. In contrast, it is possible
to develop non-standard energy conditions that are intrinsically non-linear in
the stress-energy tensor, and which exhibit much better well-controlled
behaviour when semi-classical quantum effects are introduced, at the cost of a
less direct applicability to geodesic focussing. In this article we will first
review the standard energy conditions and their various limitations. (Including
the connection to the Hawking--Ellis type I, II, III, and IV classification of
stress-energy tensors). We shall then turn to the averaged, nonlinear, and
semi-classical energy conditions, and see how much can be done once
semi-classical quantum effects are included.Comment: V1: 25 pages. Draft chapter, on which the related chapter of the book
"Wormholes, Warp Drives and Energy Conditions" (to be published by Springer),
will be based. V2: typos fixed. V3: small typo fixe
Statistical mechanics of violent relaxation in stellar systems
We discuss the statistical mechanics of violent relaxation in stellar systems following the pioneering work of Lynden-Bell (1967). The solutions of the gravitational Vlasov-Poisson system develop finer and finer filaments so that a statistical description is appropriate to smooth out the small-scales and describe the ``coarse-grained'' dynamics. In a coarse-grained sense, the system is expected to reach an equilibrium state of a Fermi-Dirac type within a few dynamical times. We describe in detail the equilibrium phase diagram and the nature of phase transitions which occur in self-gravitating systems. Then, we introduce a small-scale parametrization of the Vlasov equation and propose a set of relaxation equations for the coarse-grained dynamics. These relaxation equations, of a generalized Fokker-Planck type, are derived from a Maximum Entropy Production Principle (MEPP). We make a link with the quasilinear theory of the Vlasov-Poisson system and derive a truncated model appropriate to collisionless systems subject to tidal forces. With the aid of this kinetic theory, we qualitatively discuss the concept of ``incomplete relaxation'' and the limitations of Lynden-Bell's theory
Diffusion and scaling in escapes from two-degrees-of-freedom Hamiltonian systems
This paper summarizes an investigation of the statistical properties of
orbits escaping from three different two-degrees-of-freedom Hamiltonian
systems which exhibit global stochasticity. Each time-independent H =
H-0 + epsilon H’, with H-0 an integrable Hamiltonian and epsilon H’ a
nonintegrable correction, not necessarily small. Despite possessing very
different symmetries, ensembles of orbits in all three potentials
exhibit similar behavior. For epsilon below a critical epsilon(0),
escapes are impossible energetically. For somewhat higher values, escape
is allowed energetically but still many orbits never escape. The escape
probability P computed for an arbitrary orbit ensemble decays toward
zero exponentially. At or near a critical value epsilon(1) > epsilon(0)
there is a rather abrupt qualitative change in behavior. Above
epsilon(1), P typically exhibits (1) an initial rapid evolution toward a
nonzero P-0 (epsilon), the value of which is independent of the detailed
choice of initial conditions, followed by (2) a much slower subsequent
decay toward zero which, in at least one case, is well fit by a power
law P(t)proportional to t(-mu), with mu approximate to 0.35-0.40. In all
three cases, P-0 and the time T required to converge toward P-0 scale as
powers of epsilon-epsilon(1), i.e., P(0)proportional
to(epsilon-epsilon(1))(alpha) and T proportional
to(epsilon-epsilon(1))(beta), and T also scales in the linear size r of
the region sampled for initial conditions, i. e., T proportional to
r(-delta). To within statistical uncertainties, the best fit values of
the critical exponents alpha, beta, and delta appear to be the same for
all three potentials, namely alpha approximate to 0.5, beta approximate
to 0.4, and delta approximate to 0.1, and satisfy alpha-beta-delta
approximate to 0. The transitional behavior observed near epsilon(1) is
attributed to the breakdown of some especially significant KAM tori or
cantori. The power law behavior at late times is interpreted as
reflecting intrinsic diffusion of chaotic orbits through cantori
surrounding islands of regular orbits. (C) 1999 American Institute of
Physics. [S1054-1500(99)02302-2]