Dynamical simulations of classical stochastic systems using matrix product states
Dynamical simulations of classical stochastic systems using matrix product states
复制标题
DOI：
10.1103/physreve.82.036702
复制
发表时间：
2010-09-16
影响因子：
2.4
通讯作者：
Jaksch, D.
中科院分区：
文献类型：
--
作者：
Johnson, T. H.;Clark, S. R.;Jaksch, D.
我们采用最初设计用于模拟一维量子系统动力学的时间演化块抽取(TEBD)算法来模拟非平衡随机系统的时间演化。该方法将系统的概率分布用有限维的矩阵乘积状态(MPS)来表示，然后通过反复更新和近似重构来有效地模拟系统的时间演化。我们考察了MPS作为一种近似方法的使用，考察了将其应用于量子态向量和概率分布的解释之间的相似之处。在随机系统的背景下，我们考虑了两种用于TEBD算法的分解：非负矩阵分解(NMF)和奇异值分解(SVD)。非负矩阵分解确保近似概率分布是明显非负的。比较这些因式分解，我们发现奇异值分解的精度大大高于目前的NMF算法。然后，我们用TEBD方法模拟了具有数百个晶格点的系统的完全不对称简单排斥过程。利用TASEP稳态的精确解析结果，我们发现TEBD重现了这一状态，使得即使在通过将MPS的维度限制为非常小来严重压缩系统的描述时，计算期望值的误差也可以忽略不计。在定态之外，我们证明了对于特定的可观测，当MPS的维度增加到中等大小时，期望值收敛。
We adapt the time-evolving block decimation (TEBD) algorithm, originally devised to simulate the dynamics of one-dimensional quantum systems, to simulate the time evolution of nonequilibrium stochastic systems. We describe this method in detail; a system's probability distribution is represented by a matrix product state (MPS) of finite dimension and then its time evolution is efficiently simulated by repeatedly updating and approximately refactorizing this representation. We examine the use of MPS as an approximation method, looking at parallels between the interpretations of applying it to quantum state vectors and probability distributions. In the context of stochastic systems we consider two types of factorization for use in the TEBD algorithm: non-negative matrix factorization (NMF), which ensures that the approximate probability distribution is manifestly non-negative, and the singular value decomposition (SVD). Comparing these factorizations, we find the accuracy of the SVD to be substantially greater than current NMF algorithms. We then apply TEBD to simulate the totally asymmetric simple exclusion process (TASEP) for systems of up to hundreds of lattice sites in size. Using exact analytic results for the TASEP steady state, we find that TEBD reproduces this state such that the error in calculating expectation values can be made negligible even when severely compressing the description of the system by restricting the dimension of the MPS to be very small. Out of the steady state we show for specific observables that expectation values converge as the dimension of the MPS is increased to a moderate size.