On the dynamical behavior of the ABC model

We consider the ABC dynamics, with equal density of the three species, on the discrete ring with $N$ sites. In this case, the process is reversible with respect to a Gibbs measure with a mean field interaction that undergoes a second order phase transition. We analyze the relaxation time of the dynamics and show that at high temperature it grows at most as $N^2$ while it grows at least as $N^3$ at low temperature

Special fast diffusion with slow asymptotics. Entropy method and flow on a Riemannian manifold

We consider the asymptotic behaviour of positive solutions $u(t,x)$ of the fast diffusion equation $u_t=\Delta (u^{m}/m)={\rm div} (u^{m-1}\nabla u)$ posed for x\in\RR^d, $t>0$, with a precise value for the exponent $m=(d-4)/(d-2)$. The space dimension is $d\ge 3$ so that $m<1$, and even $m=-1$ for $d=3$. This case had been left open in the general study \cite{BBDGV} since it requires quite different functional analytic methods, due in particular to the absence of a spectral gap for the operator generating the linearized evolution. The linearization of this flow is interpreted here as the heat flow of the Laplace-Beltrami operator of a suitable Riemannian Manifold (\RR^d,{\bf g}), with a metric ${\bf g}$ which is conformal to the standard \RR^d metric. Studying the pointwise heat kernel behaviour allows to prove {suitable Gagliardo-Nirenberg} inequalities associated to the generator. Such inequalities in turn allow to study the nonlinear evolution as well, and to determine its asymptotics, which is identical to the one satisfied by the linearization. In terms of the rescaled representation, which is a nonlinear Fokker--Planck equation, the convergence rate turns out to be polynomial in time. This result is in contrast with the known exponential decay of such representation for all other values of $m$.Comment: 37 page

How hot can a heat bath get?

We study a model of two interacting Hamiltonian particles subject to a common potential in contact with two Langevin heat reservoirs: one at finite and one at infinite temperature. This is a toy model for 'extreme' non-equilibrium statistical mechanics. We provide a full picture of the long-time behaviour of such a system, including the existence/non-existence of a non-equilibrium steady state, the precise tail behaviour of the energy in such a state, as well as the speed of convergence toward the steady state. Despite its apparent simplicity, this model exhibits a surprisingly rich variety of long time behaviours, depending on the parameter regime: if the surrounding potential is 'too stiff', then no stationary state can exist. In the softer regimes, the tails of the energy in the stationary state can be either algebraic, fractional exponential, or exponential. Correspondingly, the speed of convergence to the stationary state can be either algebraic, stretched exponential, or exponential. Regarding both types of claims, we obtain matching upper and lower bounds