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 \(N^{-\eta }\) in the squared 2-Wasserstein metric, for an explicit \(\eta \in (0, 1/3]\).

我们在Kac的1D模型中引入了一个全局恒温器，该空间恒温器用于遭受二元碰撞的空间均质气体中的粒子速度，并且还与（局部）麦克斯韦恒温器相互作用。全局恒温器会重新调整所有粒子的速度，从而恢复系统的总能量，从而在相应的非线性动力学方程式中产生一个附加的漂移项。我们证明了该方程的遍历性，并表明其平衡分布的密度取决于模型的参数，可以显示出重尾巴，并且其在原点的行为范围可以从分析到\（C ^ k \ ），甚至爆炸。最后，我们证明了相关N粒子系统的混沌传播，具有均匀的阶次速率平方2-Wasserstein度量中的\（N ^ {-\ eta} \），用于显式\（\ eta \ in（0，1/3] \）。