Posters

Pedro Paulo Gondim Cardoso - Hydrodynamic behavior of long-range symmetric exclusion with a slow barrier: superdiffusive regime

We analyse the hydrodynamical behavior of the long jumps symmetric exclusion process in the presence of a slow barrier. The jump rates are given by a symmetric transition probability p(⋅) with infinite variance. When jumps occur from negative sites to non-negative ones, the rates are slowed down by a factor depending on n, α and β (with α>0 and β≥0). We obtain several partial differential equations given in terms of the regional fractional Laplacian on R∗, the set with real numbers different from zero, and with different boundary conditions. Surprisingly, in opposition to the diffusive regime, we get different regimes depending on whether α=1 (all bonds with the same rate) or α≠1.

Shiva Darshan - Numerical evidence for limiting temperature profile of an anharmonic chain undergoing periodic forcing

We present simulation results for an atom chain with harmonic pinning and quartic plus harmonic interaction undergoing random momentum flip. On the left we put a thermostat and on the right periodic forcing that scales like n^(-1/2) with the system size. In Komorowski, Lebowitz, Olla, Simon 2023, the authors conjecture that in the hydrodynamic limit the temperature profile of this chain satisfies a non-linear heat equation with non-linear Neumann boundary condition involving the limiting work done by the periodic forcing. They as well give a Green-Kubo like formula for the work done by the periodic forcing in the hydrodynamic limit. Our simulation results show that for large system sizes the numerically observed stationary temperature profile and work done by periodic forcing match closely to predicted hydrodynamic limit.

Elena Demattè - A kinetic equation describing the behavior of a gas interacting mainly with radiation

We investigate the behavior of gas molecules interacting with monochromatic radiation. We study a kinetic equation involving a Boltzmann equation coupled with a radiative transfer equation. The gas molecules in the studied kinetic model are assumed to be in a ground state or an excited state. The transition between these two states takes place either due to nonelastic collisions between two molecules in the ground state, or by the absorption and emission of photons. In the first case the transition is described by a nonlinear Boltzmann equation for the gas mixture. The evolution of the radiation density is described also by means of a kinetic equation for the photons, i.e. the radiative transfer equation in which the emission and absorption by the gas molecules are included.
We consider a scaling limit, called fast radiation limit, in which the interaction between the gas and the photons takes place much faster than the collisions between the gas molecules themselves. In the homogeneous case we show that the solutions of the limit problem solve a kinetic equation for which a well-posedness theory is considered. The proof of the convergence to the new kinetic equation is obtained analyzing the dynamics of the gas-photon system near the slow manifold of steady states.

This work is based on On a Kinetic Equation Describing the Behavior of a Gas Interacting Mainly with Radiation. Demattè, E. J Stat Phys 190, 124 (2023). https://doi.org/10.1007/s10955-023-03128-0

Eugenia Franco - Coagulation equations with source leading to anomalous self-similarity

Smoluchowski’s coagulation equation describes the evolution in time of a system of atmospheric particles, coagulating upon binary collision. In this poster we present a generalization of the classical Smoluchowski’s coagulation equation, i.e. the coagulation equation with a source term of small clusters. The source term drives the system out-of-equilibrium, leading to a rich range of different possible long-time behaviours, including anomalous self-similarity. We assume that the coagulation kernel is non gelling, homogeneous, with homogeneity γ ≤ 1, and behaves like x^(γ+λ) y^(−λ) when y ≪ x with γ+2λ ≥ 1. We argue that, when γ+2λ > 1 the long-time behaviour is self-similar and that the scaling of the self-similar solutions depends on the sign of γ + λ and on whether γ = −1 or γ < −1 or −1 < γ < 1. We present also some conjectures supporting the self-similar ansatz for the critical case γ + 2λ = 1, γ ≤ −1.

Bernhard Kepka - Longtime behavior for homoenergetic solutions in the collision dominated regime for hard potentials

We present results concerning a special class of solutions to the inhomogeneous Boltzmann equation termed homoenergetic solutions. They describe the dynamics of a dilute gas due to collisions and the action of either a shear, a dilation or a combination of both. In fact, the governing equation is a variation of the homogeneous Boltzmann equation including an additional drift term characterized by a matrix. This matrix models the presence of the shear or, respectively, the dilation. As a consequence this equation models non-equilibrium states of a dilute gas and, in contrast to the (in-)homogeneous Boltzmann equation, the property of detailed balance is lost. In our work we consider the case of hard potentials with both cutoff and non-cutoff collisions. Furthermore, we study situations in which the shear is dominant compared with the dilation. As a consequence, collisions turn out to be dominant, as can be seen using a formal Hilbert-type expansion close to a Maxwellian distribution. However, due to the action of the shear the temperature of this Maxwellian distribution goes to infinity as time tends to infinity. In addition, since collisions are dominant, the gas distribution remains close and converges to this time-dependent Maxwellian distribution. Furthermore, precise asymptotic formulas for the temperature of this limiting Maxwellian distribution can be computed via the Hilbert-type expansion.

Nicola Miele - Homoenergetic solutions for the Boltzmann-Rayleigh Equationin the Hyperbolic-Dominated Case

Homoenergetic solutions for the Boltzmann Equation are a particular class of solutions which were first introduced by Galkin and Truesdell in the 1960’s. These are a particular type of non-equilibrium solutions of the Boltzmann equation that describe the behavior of gases under mechanical deformations, e.g. shear, expansion or compression. Thus, they allow us to provide some insight on the dynamics of gases in open systems. It turns out that the long-time asymptotics of these solutions depends strongly on the balance between the collision term and the deformation term appearing in the equation. The detailed understanding of the particle distributions in the so-called hyperbolic-dominated case, i.e. when the deformation term is dominant over the collisions, is largely open and challenging.  We present a work in progress with some contributions in this direction in a linear setting, namely for the Boltzmann-Rayleigh equation.