We develop a theoretical foundation for a time-series analysis method suitable for revealing the spectrum of diffusion coefficients in mixed Brownian systems, where no prior knowledge of particle distinction is required. This method is directly relevant for particle tracking in biological systems, where diffusion processes are often non-uniform. We transform Brownian data onto the logarithmic domain, where the coefficients for individual modes of diffusion appear as distinct spectral peaks in the probability density. We refer to the method as the logarithmic measure of diffusion, or simply as the logarithmic measure. We provide a general protocol for deriving analytical expressions for the probability densities on the logarithmic domain. The protocol is applicable for any number of spatial dimensions with any number of diffusive states. The analytical form can be fitted to data to reveal multiple diffusive modes. We validate the theoretical distributions and benchmark the accuracy and sensitivity of the method by extracting multi-modal diffusion coefficients from 2D Brownian simulations of poly-disperse filament bundles. Bundling the filaments allows us to control the system non-uniformity and hence quantify the sensitivity of the method. By exploiting the anisotropy of the simulated filaments, we generalize the logarithmic measure to rotational diffusion. By fitting the analytical forms to simulation data, we confirm the method's theoretical foundation. An error analysis in the single-mode regime shows that the proposed method is comparable in accuracy to the standard mean squared displacement approach for evaluating diffusion coefficients. For the case of multi-modal diffusion, we compare the logarithmic measure against other more sophisticated methods, showing that both model selectivity and extraction accuracy are comparable for small data sets. Therefore we suggest that the logarithmic measure, as a method for multi-modal diffusion coefficient extraction, is ideally suited for small data sets, a condition often confronted in the experimental context. Finally, we critically discuss the proposed benefits of the method and its information content.

中文翻译：

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揭示多模态扩散的对数测量的基本原理

我们为适合揭示混合布朗系统中扩散系数谱的时间序列分析方法开发了理论基础，其中不需要粒子区分的先验知识。这种方法与生物系统中的粒子跟踪直接相关，其中扩散过程通常是不均匀的。我们将布朗数据转换到对数域，其中各个扩散模式的系数在概率密度中显示为不同的光谱峰。我们将该方法称为扩散的对数测量，或简称为对数测量。我们提供了一个通用协议，用于推导对数域上的概率密度的分析表达式。该协议适用于具有任意数量的扩散状态的任意数量的空间维度。分析形式可以拟合数据以揭示多种扩散模式。我们通过从多分散长丝束的 2D 布朗模拟中提取多模态扩散系数来验证理论分布并对该方法的准确性和灵敏度进行基准测试。捆绑细丝使我们能够控制系统的不均匀性，从而量化该方法的灵敏度。通过利用模拟细丝的各向异性，我们将对数测量推广到旋转扩散。通过将分析形式拟合到模拟数据，我们确定了该方法的理论基础。单模态的误差分析表明，所提出的方法在精度上与评估扩散系数的标准均方位移方法相当。对于多模态扩散的情况，我们将对数测量与其他更复杂的方法进行比较，表明模型选择性和提取精度对于小型数据集具有可比性。因此，我们建议对数测量作为多模态扩散系数提取的一种方法，非常适合小数据集，这是实验环境中经常遇到的情况。最后，我们批判性地讨论了该方法的拟议好处及其信息内容。在实验环境中经常遇到的情况。最后，我们批判性地讨论了该方法的拟议好处及其信息内容。在实验环境中经常遇到的情况。最后，我们批判性地讨论了该方法的拟议好处及其信息内容。