论文标题
通过副本方法的基质值几何布朗运动的精确熵的精确解决方案
Exact solution of free entropy for matrix-valued geometric Brownian motion with non-commutative matrices via the replica method
论文作者
论文摘要
几何布朗运动（GBM）是随机微分方程中的标准模型。在这项研究中，我们考虑了具有非共同矩阵的基质值GBM。将非交换矩阵引入矩阵值为值的GBM使得很难获得精确的解决方案，因为噪声术语的存在阻止了对角度化。但是，我们表明复制方法使我们能够克服这个困难。我们将具有非共同矩阵的矩阵值GBM的时间演化操作员映射到量子自旋系统中使用的各向同性Lipkin-meshkov-Glick模型的分区函数中。然后，解决各向同性Lipkin-meshkov-Glick模型的特征值问题，我们获得了自由熵的分析表达。数值模拟与我们的分析结果一致。因此，我们的表达是具有非共同基质的基质值GBM的自由熵的精确解。
Geometric Brownian motion (GBM) is a standard model in stochastic differential equations. In this study, we consider a matrix-valued GBM with non-commutative matrices. Introduction of non-commutative matrices into the matrix-valued GBM makes it difficult to obtain an exact solution because the existence of noise terms prevents diagonalization. However, we show that the replica method enables us to overcome this difficulty. We map the trace of the time evolution operator of the matrix-valued GBM with non-commutative matrices into the partition function of the isotropic Lipkin-Meshkov-Glick model used in quantum spin systems. Then, solving the eigenvalue problem of the isotropic Lipkin-Meshkov-Glick model, we obtain an analytical expression of the free entropy. Numerical simulation is consistent with our analytical result. Thus, our expression is the exact solution of the free entropy for the matrix-valued GBM with non-commutative matrices.
