论文标题
与混合哈密顿和非汉密尔顿相互作用的远程系统中的放松动力学
Relaxation dynamics in a long-range system with mixed Hamiltonian and non-Hamiltonian interactions
论文作者
论文摘要
有时，物理系统的动力学是通过非汉密尔顿运动方程来描述的，此外，该系统的特征是远程相互作用。一个具体的例子是在自由电子激光和冷原子实验中遇到的光相互作用的颗粒。在这项工作中，我们更精确地研究了非哈米尔顿体系的放松动力学，以与哈密顿量和非哈米尔顿统治相互作用的系统。我们的型号由$ n $全球耦合的粒子组成，在单位半径圆上移动；该模型是一维的。我们表明，在无限尺寸的极限中，弗拉索夫方程式描述了与哈密顿情况相似的动力学。在哈密顿案中，该系统最终达到了平衡状态，即使人们必须等待很长时间以$ n $的差异才能发生。相比之下，在非汉密尔顿病例中，没有均衡状态预计该系统最终将达到。我们用平均磁化表征了这种状态。我们发现，放松动力学在很大程度上取决于哈密顿量和非汉顿对相互作用的贡献的相对重量。当非汉顿部分主要是主导的时，磁化值将达到消失的值，这表明该系统不会以固定或旋转为恒定的磁化状态维持状态。另一方面，当哈密顿部分主要是主导的时，磁化会带来长期的强振荡，我们为此提供了启发式的解释。此外，我们发现有限尺寸的更正比在哈密顿案中的校正要明显得多。我们通过证明Lenard-Balescu方程对Vlasov方程进行了领先的校正并不消失，这与一维哈密顿长期远程系统中发生的情况相反，这证明了这一点。
Sometimes the dynamics of a physical system is described by non-Hamiltonian equations of motion, and additionally, the system is characterized by long-range interactions. A concrete example is that of particles interacting with light as encountered in free-electron laser and cold-atom experiments. In this work, we study the relaxation dynamics to non-Hamiltonian systems, more precisely, to systems with interactions of both Hamiltonian and non-Hamiltonian origin. Our model consists of $N$ globally-coupled particles moving on a circle of unit radius; the model is one-dimensional. We show that in the infinite-size limit, the dynamics, similarly to the Hamiltonian case, is described by the Vlasov equation. In the Hamiltonian case, the system eventually reaches an equilibrium state, even though one has to wait for a long time diverging with $N$ for this to happen. By contrast, in the non-Hamiltonian case, there is no equilibrium state that the system is expected to reach eventually. We characterize this state with its average magnetization. We find that the relaxation dynamics depends strongly on the relative weight of the Hamiltonian and non-Hamiltonian contributions to the interaction. When the non-Hamiltonian part is predominant, the magnetization attains a vanishing value, suggesting that the system does not sustain states with constant magnetization, either stationary or rotating. On the other hand, when the Hamiltonian part is predominant, the magnetization presents long-lived strong oscillations, for which we provide a heuristic explanation. Furthermore, we find that the finite-size corrections are much more pronounced than those in the Hamiltonian case; we justify this by showing that the Lenard-Balescu equation, which gives leading-order corrections to the Vlasov equation, does not vanish, contrary to what occurs in one-dimensional Hamiltonian long-range systems.
