20 research outputs found
On Strong Convergence to Equilibrium for the Boltzmann Equation with Soft Potentials
The paper concerns - convergence to equilibrium for weak solutions of
the spatially homogeneous Boltzmann Equation for soft potentials (-4\le
\gm<0), with and without angular cutoff. We prove the time-averaged
-convergence to equilibrium for all weak solutions whose initial data have
finite entropy and finite moments up to order greater than 2+|\gm|. For the
usual -convergence we prove that the convergence rate can be controlled
from below by the initial energy tails, and hence, for initial data with long
energy tails, the convergence can be arbitrarily slow. We also show that under
the integrable angular cutoff on the collision kernel with -1\le \gm<0, there
are algebraic upper and lower bounds on the rate of -convergence to
equilibrium. Our methods of proof are based on entropy inequalities and moment
estimates.Comment: This version contains a strengthened theorem 3, on rate of
convergence, considerably relaxing the hypotheses on the initial data, and
introducing a new method for avoiding use of poitwise lower bounds in
applications of entropy production to convergence problem
Phase transitions towards frequency entrainment in large oscillator lattices
We investigate phase transitions towards frequency entrainment in large,
locally coupled networks of limit cycle oscillators. Specifically, we simulate
two-dimensional lattices of pulse-coupled oscillators with random natural
frequencies, resembling pacemaker cells in the heart. As coupling increases,
the system seems to undergo two phasetransitions in the thermodynamic limit. At
the first, the largest cluster of frequency entrained oscillators becomes
macroscopic. At the second, global entrainment settles. Between the two
transitions, the system has features indicating self-organized criticality.Comment: 4 pages, 5 figures, submitted to PR
Collisional rates for the inelastic Maxwell model: application to the divergence of anisotropic high-order velocity moments in the homogeneous cooling state
The collisional rates associated with the isotropic velocity moments
and
are exactly derived in the case of the
inelastic Maxwell model as functions of the exponent , the coefficient of
restitution , and the dimensionality . The results are applied to
the evolution of the moments in the homogeneous free cooling state. It is found
that, at a given value of , not only the isotropic moments of a degree
higher than a certain value diverge but also the anisotropic moments do. This
implies that, while the scaled distribution function has been proven in the
literature to converge to the isotropic self-similar solution in well-defined
mathematical terms, nonzero initial anisotropic moments do not decay with time.
On the other hand, our results show that the ratio between an anisotropic
moment and the isotropic moment of the same degree tends to zero.Comment: 7 pages, 2 figures; v2: clarification of some mathematical statements
and addition of 7 new references; v3: Published in "Special Issue: Isaac
Goldhirsch - A Pioneer of Granular Matter Theory
'It's Complicated': Canadian Correctional Officer Recruits’ Interpretations of Issues Relating to the Presence of Transgender Prisoners
Drawing upon semi-structured interviews with correctional officer recruits in Q2 training (n = 55), we reflect on recruit interpretations of transgender (trans) prisoner placement within federal prisons in light of recent changes instigated by Canadian Prime Minister Trudeau. Recognising that prison is a carceral and gender binary space, we assert that trans prisoner lives and experiences cannot easily be appropriately recognised or included in prison policy and prisoner management procedures. Our findings reveal that most recruits are supportive and appreciative of the complexities of trans experiences, yet some, especially those with prior experience working in prisons, describe occupational strains tied to accommodating trans prisoners