论文标题
Matsubara动力学与精确量子动力学的振荡器的比较
Comparison of Matsubara dynamics with exact quantum dynamics for an oscillator coupled to a dissipative bath
论文作者
论文摘要
Matsubara Dynamics是经典的动力学，当假想的时间路径综合分析平滑时会产生。它保存了量子螺栓分布，并以途径近似形式出现在路径综合动力学方法中，例如（恒温）环聚合物分子动力学（T）RPMD和质心分子动力学（CMD）。但是，从未将其与非线性操作员的精确量子动力学直接进行比较，因为治疗阶段的难度将Matsubara模式的数量限制在少于10中。在这里，我们将多达$ \ sim $ \ sim $ \ sim $ 200 Matsubara模式用于模拟振动器的模拟中，偶然振荡了和散发的和谐振动器的耗散浴缸。这是通过以广义Langevin方程的形式表达Matsubara运动方程，近似于真实的噪声，并在分析上继续进行MONMA，以将Matsubara相转换为环形聚合物弹簧。所得的运动方程稳定至最大值的模式值，随着浴耦合强度而增加，并随着系统的非谐度而降低。发现高度振荡的Matsubara模式的尾巴动力学是谐波的，因此可以有效地计算出来。对于中度的无谐振荡器，与浴缸具有强大但亚临界的耦合，Matsubara模拟产生了非线性$ \ ligal \ langle {\ hat q^2 \ hat q^2（t）} \ time $ thime \ time $ time $ time pime-time prolaction几乎完全与确切的量子量相吻合。对于较弱的耦合强度，还获得了合理的一致性，在这种情况下，由于近似值而出现错误。这些结果提供了有力的证据，表明Matsubara动力学正确地解释了在热平衡中的量子系统中如何产生经典动力学。
Matsubara dynamics is the classical dynamics which results when imaginary-time path-integrals are smoothed; it conserves the quantum Boltzmann distribution and appears in drastically approximated form in path-integral dynamics methods such as (thermostatted) ring-polymer molecular dynamics (T)RPMD and centroid molecular dynamics (CMD). However, it has never been compared directly with exact quantum dynamics for non-linear operators, because the difficulty of treating the phase has limited the number of Matsubara modes to fewer than 10. Here, we treat up to $\sim$200 Matsubara modes in simulations of a Morse oscillator coupled to a dissipative bath of harmonic oscillators. This is done by expressing the Matsubara equations of motion in the form of a generalised Langevin equation, approximating the noise to be real, and analytically continuing the momenta to convert the Matsubara phase into ring-polymer springs. The resulting equations of motion are stable up to a maximum value of modes which increases with bath coupling strength and decreases with system anharmonicity. The dynamics of the tail of highly oscillatory Matsubara modes is found to be harmonic, and can thus be computed efficiently. For a moderately anharmonic oscillator with a strong but subcritical coupling to the bath, the Matsubara simulations yield non-linear $\large\langle{\hat q^2\hat q^2(t)}\large\rangle$ time-correlation functions in almost perfect agreement with the exact quantum results. Reasonable agreement is also obtained for weaker coupling strengths, where errors arise because of the real-noise approximation. These results give strong evidence that Matsubara dynamics correctly explains how classical dynamics arises in quantum systems which are in thermal equilibrium.
