6,921 research outputs found
Wave-Function renormalization and the Hopf algebra of Connes and Kreimer
In this talk, we show how the Connes-Kreimer Hopf algebra morphism can be
extended when taking into account the wave-function renormalization. This leads
us to a semi-direct product of invertible power series by formal
diffeomorphisms.Comment: 5 pages, no figure, talk presented in the conference "Brane New World
and Noncommutative Geometry", Torino, Villa Gualino,(Italy) Octobe
Models of dark matter halos based on statistical mechanics: I. The classical King model
We consider the possibility that dark matter halos are described by the
Fermi-Dirac distribution at finite temperature. This is the case if dark matter
is a self-gravitating quantum gas made of massive neutrinos at statistical
equilibrium. This is also the case if dark matter can be treated as a
self-gravitating collisionless gas experiencing Lynden-Bell's type of violent
relaxation. In order to avoid the infinite mass problem and carry out a
rigorous stability analysis, we consider the fermionic King model. In this
paper, we study the non-degenerate limit leading to the classical King model.
This model was initially introduced to describe globular clusters. We propose
to apply it also to large dark matter halos where quantum effects are
negligible. We determine the caloric curve and study the thermodynamical
stability of the different configurations. Equilibrium states exist only above
a critical energy in the microcanonical ensemble and only above a
critical temperature in the canonical ensemble. For , the system
undergoes a gravothermal catastrophe and, for , it undergoes an
isothermal collapse. We compute the profiles of density, circular velocity, and
velocity dispersion. We compare the prediction of the classical King model to
the observations of large dark matter halos. Because of collisions and
evaporation, the central density increases while the slope of the halo density
profile decreases until an instability takes place. We show that large dark
matter halos are relatively well-described by the King model at, or close to,
the point of marginal microcanonical stability. At that point, the King model
generates a density profile that can be approximated by the modified Hubble
profile. This profile has a flat core and decreases as at large
distances, like the observational Burkert profile. Less steep halos are
unstable
Models of dark matter halos based on statistical mechanics: II. The fermionic King model
We discuss the nature of phase transitions in the fermionic King model which
describes tidally truncated quantum self-gravitating systems. This distribution
function takes into account the escape of high energy particles and has a
finite mass. On the other hand, the Pauli exclusion principle puts an upper
bound on the phase space density of the system and stabilizes it against
gravitational collapse. As a result, there exists a statistical equilibrium
state for any accessible values of energy and temperature. We plot the caloric
curves and investigate the nature of phase transitions as a function of the
degeneracy parameter in both microcanonical and canonical ensembles. We
consider stable and metastable states and emphasize the importance of the
latter for systems with long-range interactions. Phase transitions can take
place between a "gaseous" phase unaffected by quantum mechanics and a
"condensed" phase dominated by quantum mechanics. The phase diagram exhibits
two critical points, one in each ensemble, beyond which the phase transitions
disappear. There also exist a region of negative specific heats and a situation
of ensemble inequivalence for sufficiently large systems. We apply the
fermionic King model to the case of dark matter halos made of massive
neutrinos. The gaseous phase describes large halos and the condensed phase
describes dwarf halos. Partially degenerate configurations describe
intermediate size halos. We argue that large dark matter halos cannot harbor a
fermion ball because these nucleus-halo configurations are thermodynamically
unstable (saddle points of entropy). Large dark matter halos may rather contain
a central black hole resulting from a dynamical instability of relativistic
origin occurring during the gravothermal catastrophe
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