Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, 606-8501, Kyoto, Japan.
Phys Rev E. 2017 May;95(5-1):052148. doi: 10.1103/PhysRevE.95.052148. Epub 2017 May 30.
The linear response is investigated in a long-range Hamiltonian system from the viewpoint of dynamics, which is described by the Vlasov equation in the large-population limit. Because of the existence of the Casimir invariants of the Vlasov dynamics, an external field does not drive the system to the forced thermal equilibrium in general, and the linear response is suppressed. With the aid of a linear response theory based on the Vlasov dynamics, we compute the suppressed linear response in a system having two order parameters, which introduce the conjugate two external fields and the susceptibility matrix of size 2 accordingly. Moreover, the two order parameters bring three phases and there are three types of second-order phase transitions between them. For each type of phase transition, all the critical exponents for elements of the susceptibility matrix are computed. The critical exponents reveal that some elements of the matrices do not diverge even at critical points, while the mean-field theory predicts divergences. The linear response theory also suggests the appearance of negative off-diagonal elements; in other words, an applied external field decreases the value of an order parameter. These theoretical predictions are confirmed by direct numerical simulations of the Vlasov equation.
从动力学的角度研究了长程哈密顿系统中的线性响应，该系统由大种群极限下的 Vlasov 方程描述。由于 Vlasov 动力学的 Casimir 不变量的存在，一般来说，外部场不会使系统达到强制热平衡，从而抑制了线性响应。借助基于 Vlasov 动力学的线性响应理论，我们计算了具有两个序参量的系统中的抑制线性响应，这相应地引入了两个共轭外部场和大小为 2 的磁化率矩阵。此外，这两个序参量带来了三个相，它们之间存在三种类型的二级相变。对于每种类型的相变，我们都计算了磁化率矩阵元素的所有临界指数。临界指数表明，即使在临界点，矩阵的某些元素也不会发散，而平均场理论则预测会发散。线性响应理论还表明出现了负非对角元素；换句话说，施加的外部场会降低一个序参量的值。这些理论预测得到了 Vlasov 方程的直接数值模拟的证实。
