Department of Chemistry, Physical Chemistry, University of Gothenburg, SE-412-96 Gothenburg, Sweden.
J Chem Phys. 2011 Jan 21;134(3):034118. doi: 10.1063/1.3519637.
We consider the Dirac-Frenkel variational principle in Wigner phase-space and apply it to the Wigner-Liouville equation for both imaginary and real time dynamical problems. The variational principle allows us to deduce the optimal time-evolution of the parameter-dependent Wigner distribution. It is shown that the variational principle can be formulated alternatively as a "principle of least action." Several low-dimensional problems are considered. In imaginary time, high-temperature classical distributions are "cooled" to arrive at low-temperature quantum Wigner distributions whereas in real time, the coherent dynamics of a particle in a double well is considered. Especially appealing is the relative ease at which Feynman's path integral centroid variable can be incorporated as a variational parameter. This is done by splitting the high-temperature Boltzmann distribution into exact local centroid constrained distributions, which are thereafter cooled using the variational principle. The local distributions are sampled by Metropolis Monte Carlo by performing a random walk in the centroid variable. The combination of a Monte Carlo and a variational procedure enables the study of quantum effects in low-temperature many-body systems, via a method that can be systematically improved.
我们考虑狄拉克-芬克尔变分原理在维格纳相空间，并将其应用于虚时间和实时间动力学问题的维格纳-刘维尔方程。变分原理允许我们推导出参数依赖的维格纳分布的最优时间演化。结果表明，变分原理可以被表述为一种“最小作用量原理”。我们考虑了几个低维问题。在虚时间中，高温经典分布被“冷却”到低温量子维格纳分布，而在实时间中，我们考虑了一个粒子在双势阱中的相干动力学。特别吸引人的是，费曼路径积分质心变量可以很容易地作为一个变分参数被包含进去。这是通过将高温玻尔兹曼分布分解为精确的局部质心约束分布来实现的，然后通过变分原理对其进行冷却。通过在质心变量中进行随机游走，我们使用蒙特卡罗方法对局部分布进行抽样。蒙特卡罗和变分方法的结合，使得我们可以通过一种可以系统改进的方法，研究低温多体系统中的量子效应。
