Boltzmann-conserving classical dynamics in quantum time-correlation functions: "Matsubara dynamics"
Boltzmann-conserving classical dynamics in quantum time-correlation functions: "Matsubara dynamics"
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DOI：
10.1063/1.4916311
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发表时间：
2015-04-07
影响因子：
4.4
通讯作者：
Althorpe, Stuart C.
中科院分区：
文献类型：
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作者：
Hele, Timothy J. H.;Willatt, Michael J.;Althorpe, Stuart C.
我们表明，在线性化的半经典初始值表示（LSC-IVR或“经典Wigner近似”）的推导中，一个单一的变化会导致经典动力学守恒量子玻尔兹曼分布。我们通过根据自由环聚合物的正常模式（即离散虚时间费曼路径）编写（精确）量子时间相关函数来重新推导（标准）LSC-IVR方法，取聚合物珠数N ->无穷大的极限，使得最低正常模式频率取其“Matsubara”值。我们提出的改变是截断量子Liouvillian，不是明确地以(h) / bar(2)在(h) / bar(0)的幂（这给出了标准的LSC-IVR近似），而是以对应于最低Matsubara频率的正模导数。由此产生的“Matsubara”动力学本质上是经典的(因为所有项O（（h) / bar(2)）在极限N ->∞中从Matsubara Liouvillian中消失），并且守恒量子玻尔兹曼分布，因为Matsubara哈密顿量相对于虚时平移是对称的。数值实验表明，量子时间相关函数的Matsubara近似在模数上收敛，并且比LSC-IVR更符合精确的量子结果。Matsubara动力学的计算成本太高，不能应用于复杂系统，但它的进一步逼近可能导致实用的方法。(C) 2015 AIP出版有限责任公司
We show that a single change in the derivation of the linearized semiclassical-initial value representation (LSC-IVR or "classical Wigner approximation") results in a classical dynamics which conserves the quantum Boltzmann distribution. We rederive the (standard) LSC-IVR approach by writing the (exact) quantum time-correlation function in terms of the normal modes of a free ring-polymer (i.e., a discrete imaginary-time Feynman path), taking the limit that the number of polymer beads N -> infinity, such that the lowest normal-mode frequencies take their "Matsubara" values. The change we propose is to truncate the quantum Liouvillian, not explicitly in powers of (h) over bar (2) at (h) over bar (0) (which gives back the standard LSC-IVR approximation), but in the normal-mode derivatives corresponding to the lowest Matsubara frequencies. The resulting "Matsubara" dynamics is inherently classical (since all terms O((h) over bar (2)) disappear from the Matsubara Liouvillian in the limit N -> infinity) and conserves the quantum Boltzmann distribution because the Matsubara Hamiltonian is symmetric with respect to imaginary-time translation. Numerical tests show that the Matsubara approximation to the quantum time-correlation function converges with respect to the number of modes and gives better agreement than LSC-IVR with the exact quantum result. Matsubara dynamics is too computationally expensive to be applied to complex systems, but its further approximation may lead to practical methods. (C) 2015 AIP Publishing LLC.