Fricks John, Yao Lingxing, Elston Timothy C, Gregory Forest And M
Department of Statistics, Penn State University, University Park, PA 16802.
Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599-3250 ; Department of Pharmacology, University of North Carolina, Chapel Hill, NC 27599-7365.
SIAM J Appl Math. 2009;69(5):1277-1308. doi: 10.1137/070695186.
Passive microrheology [12] utilizes measurements of noisy, entropic fluctuations (i.e., diffusive properties) of micron-scale spheres in soft matter to infer bulk frequency-dependent loss and storage moduli. Here, we are concerned exclusively with diffusion of Brownian particles in viscoelastic media, for which the Mason-Weitz theoretical-experimental protocol is ideal, and the more challenging inference of bulk viscoelastic moduli is decoupled. The diffusive theory begins with a generalized Langevin equation (GLE) with a memory drag law specified by a kernel [7, 16, 22, 23]. We start with a discrete formulation of the GLE as an autoregressive stochastic process governing microbead paths measured by particle tracking. For the inverse problem (recovery of the memory kernel from experimental data) we apply time series analysis (maximum likelihood estimators via the Kalman filter) directly to bead position data, an alternative to formulas based on mean-squared displacement statistics in frequency space. For direct modeling, we present statistically exact GLE algorithms for individual particle paths as well as statistical correlations for displacement and velocity. Our time-domain methods rest upon a generalization of well-known results for a single-mode exponential kernel [1, 7, 22, 23] to an arbitrary -mode exponential series, for which the GLE is transformed to a vector Ornstein-Uhlenbeck process.
被动式微观流变学[12]利用对软物质中微米级球体的噪声熵涨落（即扩散特性）进行测量，来推断与频率相关的体积损耗模量和储能模量。在此，我们仅关注布朗粒子在粘弹性介质中的扩散，对此梅森 - 韦茨理论 - 实验方案是理想的，并且体积粘弹性模量这一更具挑战性的推断被解耦。扩散理论始于一个广义朗之万方程（GLE），其具有由一个核指定的记忆拖曳定律[7, 16, 22, 23]。我们从将GLE离散化为一个自回归随机过程开始，该过程用于控制通过粒子追踪测量的微珠路径。对于反问题（从实验数据中恢复记忆核），我们将时间序列分析（通过卡尔曼滤波器的最大似然估计器）直接应用于珠子位置数据，这是一种替代基于频率空间中均方位移统计的公式的方法。对于直接建模，我们给出了单个粒子路径的统计精确GLE算法以及位移和速度的统计相关性。我们的时域方法基于将单模指数核[1, 7, 22, 23]的著名结果推广到任意模指数级数，对于该级数，GLE被转换为向量奥恩斯坦 - 乌伦贝克过程。
