13,172 research outputs found
Gravitational phase transitions with an exclusion constraint in position space
We discuss the statistical mechanics of a system of self-gravitating
particles with an exclusion constraint in position space in a space of
dimension . The exclusion constraint puts an upper bound on the density of
the system and can stabilize it against gravitational collapse. We plot the
caloric curves giving the temperature as a function of the energy and
investigate the nature of phase transitions as a function of the size of the
system and of the dimension of space 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. For , there is no phase transition. For , phase transitions can take place
between a "gaseous" phase unaffected by the exclusion constraint and a
"condensed" phase dominated by this constraint. The condensed configurations
have a core-halo structure made of a "rocky core" surrounded by an
"atmosphere", similar to a giant gaseous planet. For large systems there exist
microcanonical and canonical first order phase transitions. For intermediate
systems, only canonical first order phase transitions are present. For small
systems there is no phase transition at all. As a result, the phase diagram
exhibits two critical points, one in each ensemble. There also exist a region
of negative specific heats and a situation of ensemble inequivalence for
sufficiently large systems. By a proper interpretation of the parameters, our
results have application for the chemotaxis of bacterial populations in biology
described by a generalized Keller-Segel model including an exclusion constraint
in position space. They also describe colloids at a fluid interface driven by
attractive capillary interactions when there is an excluded volume around the
particles. Connexions with two-dimensional turbulence are also mentioned
A relaxation model for liquid-vapor phase change with metastability
We propose a model that describes phase transition including meta\-stable
states present in the van der Waals Equation of State. From a convex
optimization problem on the Helmoltz free energy of a mixture, we deduce a
dynamical system that is able to depict the mass transfer between two phases,
for which equilibrium states are either metastable states, stable states or {a
coexistent state}. The dynamical system is then used as a relaxation source
term in an isothermal 44 two-phase model. We use a Finite Volume scheme
that treats the convective part and the source term in a fractional step way.
Numerical results illustrate the ability of the model to capture phase
transition and metastable states
Metastability of Asymptotically Well-Behaved Potential Games
One of the main criticisms to game theory concerns the assumption of full
rationality. Logit dynamics is a decentralized algorithm in which a level of
irrationality (a.k.a. "noise") is introduced in players' behavior. In this
context, the solution concept of interest becomes the logit equilibrium, as
opposed to Nash equilibria. Logit equilibria are distributions over strategy
profiles that possess several nice properties, including existence and
uniqueness. However, there are games in which their computation may take time
exponential in the number of players. We therefore look at an approximate
version of logit equilibria, called metastable distributions, introduced by
Auletta et al. [SODA 2012]. These are distributions that remain stable (i.e.,
players do not go too far from it) for a super-polynomial number of steps
(rather than forever, as for logit equilibria). The hope is that these
distributions exist and can be reached quickly by logit dynamics.
We identify a class of potential games, called asymptotically well-behaved,
for which the behavior of the logit dynamics is not chaotic as the number of
players increases so to guarantee meaningful asymptotic results. We prove that
any such game admits distributions which are metastable no matter the level of
noise present in the system, and the starting profile of the dynamics. These
distributions can be quickly reached if the rationality level is not too big
when compared to the inverse of the maximum difference in potential. Our proofs
build on results which may be of independent interest, including some spectral
characterizations of the transition matrix defined by logit dynamics for
generic games and the relationship of several convergence measures for Markov
chains
Newtonian gravity in d dimensions
We study the influence of the dimension of space on the thermodynamics of the
classical and quantum self-gravitating gas. We consider Hamiltonian systems of
self-gravitating particles described by the microcanonical ensemble and
self-gravitating Brownian particles described by the canonical ensemble. We
present a gallery of caloric curves in different dimensions of space and
discuss the nature of phase transitions as a function of the dimension d. We
also provide the general form of the Virial theorem in d dimensions and discuss
the particularity of the dimension d=4 for Hamiltonian systems and the
dimension d=2 for Brownian systems
Phase transitions in self-gravitating systems and bacterial populations with a screened attractive potential
We consider a system of particles interacting via a screened Newtonian
potential and study phase transitions between homogeneous and inhomogeneous
states in the microcanonical and canonical ensembles. Like for other systems
with long-range interactions, we obtain a great diversity of microcanonical and
canonical phase transitions depending on the dimension of space and on the
importance of the screening length. We also consider a system of particles in
Newtonian interaction in the presence of a ``neutralizing background''. By a
proper interpretation of the parameters, our study describes (i)
self-gravitating systems in a cosmological setting, and (ii) chemotaxis of
bacterial populations in the original Keller-Segel model
- âŠ