一类带重标度的恒温Kac模型
On a thermostated Kac model with rescaling
摘要：
我们介绍了一个全球恒温器卡茨的一维模型的粒子在空间均匀的气体进行二进制碰撞的速度，也与（本地）麦克斯韦恒温器。全局恒温器重新调整所有粒子的速度，从而恢复系统的总能量，这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的遍历性，并表明其平衡分布的密度，根据模型的参数，可以表现出沉重的尾巴，其行为在原点的范围可以从分析，是$C^k$，甚至爆破。最后，我们证明了相关的$N$ -粒子系统的混沌传播，在平方$2$ -Wasserstein度量的顺序$N^{-η}$在时间上均匀的速率，明确的$η\in (0, 1/3]$。
英文摘要：
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 $η\in (0, 1/3]$.
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