Makiko Sasada

Hydrodynamic Limit for an Evolutional Model of Two-Dimensional Young Diagrams

We construct dynamics of two-dimensional Young diagrams, which are naturally associated with their grandcanonical ensembles, by allowing the creation and annihilation of unit squares located at the boundary of the diagrams. The grandcanonical ensembles, which were introduced by Vershik [17], are uniform measures under conditioning on their size (or equivalently, area). We then show that, as the averaged size of the diagrams diverges, the corresponding height variable converges to a solution of a certain non-linear partial differential equation under a proper hydrodynamic scaling. Furthermore, the stationary solution of the limit equation is identified with the so-called Vershik curve. We discuss both uniform and restricted uniform statistics for the Young diagrams.

Macroscopic energy diffusion for a chain of anharmonic oscillators

We study the energy diffusion in a chain of anharmonic oscillators where the Hamiltonian dynamics is perturbed by a local energy conserving noise. We prove that under diffusive rescaling of space–time, energy fluctuations diffuse and evolve following an infinite dimensional linear stochastic differential equation driven by the linearized heat equation. We also give variational expressions for the thermal diffusivity and some upper and lower bounds.

Fluctuations in an evolutional model of two-dimensional Young diagrams

We discuss the non-equilibrium fluctuation problem, which corresponds to the hydrodynamic limit established by Funaki and Sasada (2010) [9], for the dynamics of two-dimensional Young diagrams associated with the uniform and restricted uniform statistics, and derive linear stochastic partial differential equations in the limit. We show that their invariant measures are identical to the Gaussian measures which appear in the fluctuation limits in the static situations.

Spectral gap for stochastic energy exchange model with nonuniformly positive rate function

We give a lower bound on the spectral gap for a class of stochastic energy exchange models. In 2011, Grigo et al. introduced the model and showed that, for a class of stochastic energy exchange models with a uniformly positive rate function, the spectral gap of an N-component system is bounded from below by a function of order N−2. In this paper, we consider the case where the rate function is not uniformly positive. For this case, the spectral gap depends not only on N but also on the averaged energy E, which is the conserved quantity under the dynamics. Under some assumption, we obtain a lower bound of the spectral gap which is of order C(E)N−2 where C(E) is a positive constant depending on E. As a corollary of the result, a lower bound of the spectral gap for the mesoscopic energy exchange process of billiard lattice studied by Gaspard and Gilbert [ J. Stat. Mech. Theory Exp. 2008 (2008) p11021, J. Stat. Mech. Theory Exp. 2009 (2009) p08020] and the stick process studied by Feng et al. [ Stochastic Process. Appl. 66 (1997) 147–182] are obtained.

From Normal Diffusion to Superdiffusion of Energy in the Evanescent Flip Noise Limit

We consider a harmonic chain perturbed by a stochastic noise which conserves the energy and a second quantity called the volume, and destroys all the other ones. We then add to this model a second energy conserving noise depending on a parameter

On the Spectral Gap of the Kac Walk and Other Binary Collision Processes on d-Dimensional Lattice

\gamma

Thermal Conductivity for Coupled Charged Harmonic Oscillators with Noise in a Magnetic Field

γ, that annihilates the volume conservation. When