## On Kac’s model, ideal Thermostats, and finite Reservoirs

Abstract: In 1956, Mark Kac introduced a stochastic model to derive a Boltzmann-like equation. Like the space-homogeneous Boltzmann’s equation, Kac’s equation is ergodic with centered Gaussians as the unique equilibrium state. In this talk, I will introduce Kac’s model, the thermostat used in [1] to guarantee exponentially fast convergence to equilibrium, and sketch the result in [2] how this infinite thermostat can be approximated by a finite but large reservoir. References: [1]Bonetto, F., Loss, M.,Vaidyanathan, R.: J. Stat. Phys. 156(4), 647– 667 (2014) [2]Bonetto...

Read More## Scaling limits for a slowed random walk driven by symmetric exclusion

Abstract: Consider a simple symmetric exclusion process in one dimension, and a random walk on the same space. When on top of particles, the walker has a drift to the left, when on top of holes it has a drift to the right. Under weakly asymmetric scaling, we prove a law of large numbers and a functional central limit theorem for the position of this random walk. The proof uses techniques from the field of hydrodynamic limits to study the fluctuations of the number of particles of the in large boxes around the walker.

Read More## Particle systems and propagation of chaos for some kinetic models

Abstract: In this talk we will make a quick historical review of some equations arising in the classical kinetic theory of gases and related models. We will start with the Boltzmann equation, which describes the evolution of the distribution of positions and velocities of infinitely many small particles of a gas in 3-dimensional space, subjected to elastic binary collisions. We consider a finite $N$-particle system and introduce the important concept of propagation of chaos: the convergence, as $N\to\infty$ and for each time $t\geq 0$, of the distribution of the particles towards the...

Read More## Two-time distribution for KPZ growth in one dimension

Abstract: Consider the height fluctuations H(x,t) at spatial point x and time t of one-dimensional growth models in the Kardar-Parisi-Zhang (KPZ) class. The spatial point process at a single time is known to converge at large time to the Airy processes (depending on the initial data). The multi-time process however is less well understood. In this talk, I will discuss the result by Johansson on the two-time problem, namely the joint distribution of (H(x,t),H(x,at)) with a>0, in the case of droplet initial data. I also show how to adapt his approach to the flat initial case. This is...

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