论文标题
在带有续订的恒温KAC模型上
On a thermostated Kac model with rescaling
论文作者
论文摘要
我们引入了有关KAC的1D模型的全局恒温器，该模型在经受二进制碰撞的空间同质气体中的颗粒速度，也与（局部）Maxwellian恒温器相互作用。全局恒温器重新缩放了所有颗粒的速度，从而恢复了系统的总能量，从而导致相应的非线性动力学方程中的额外漂移项。我们证明了这个方程式的奇特性，并表明其平衡分布的密度取决于模型的参数，可以表现出沉重的尾巴，其在原点的行为可以从分析到$ c^k $，甚至甚至是爆炸。最后，我们证明了相关的$ n $零件系统的混乱繁殖，并在平方$ 2 $ -WASSERSTEIN METRIC中订单$ n^{ - η} $均匀的订单率，用于显式$η\ in（0，1/3] $。
We introduce a global thermostat on Kac's 1D model for the velocities of particles in a space-homogeneous gas subjected to binary collisions, also interacting with a (local) Maxwellian thermostat. The global thermostat rescales the velocities of all the particles, thus restoring the total energy of the system, which leads to an additional drift term in the corresponding nonlinear kinetic equation. We prove ergodicity for this equation, and show that its equilibrium distribution has a density that, depending on the parameters of the model, can exhibit heavy tails, and whose behaviour at the origin can range from being analytic, to being $C^k$, and even to blowing-up. Finally, we prove propagation of chaos for the associated $N$-particle system, with a uniform-in-time rate of order $N^{-η}$ in the squared $2$-Wasserstein metric, for an explicit $η\in (0, 1/3]$.
