We studied the dissipation effect induced by the internal relative motion between two nonlinear $\sigma$-models (NLSMs). The internal degrees of freedom (DOF) of the two NLSMs coupled to each other via an interface potential. We performed the path integral of the vacuum persistence amplitude (VPA) and got the quantum action. Then we introduced the relative motion via a boost transformation with velocity parameter $v=|v|$. Furthermore, we calculated the imaginary part of the boosted quantum action which was proportional to the vacuum decay probability. At last, we calculated the frictional force. The numerical results gave the velocity $v$ dependency of the vacuum decay probability, and the velocity $v$ dependency of the frictional force. There was a threshold of relative velocity $v$ for the vacuum decay probability and the frictional force. We gave a classical argument of the existence of the threshold, and the argument matched the mathematical results.

我们研究了由两个非线性之间的内部相对运动引起的耗散效应$\sigma$-模型（NLSM）。两个 NLSM 通过界面电位相互耦合的内部自由度 (DOF)。我们进行了真空持续振幅（VPA）的路径积分，得到了量子作用。然后我们通过带有速度参数的boost变换引入了相对运动$v=|v|$. 此外，我们计算了与真空衰减概率成正比的增强量子作用的虚部。最后，我们计算了摩擦力。数值结果给出了速度$v$真空衰减概率和速度的相关性$v$摩擦力的依赖性。存在相对速度阈值$v$为真空衰减概率和摩擦力。我们给出了阈值存在的经典论证，该论证与数学结果相匹配。