This article explores the mathematical description of anomalous diffusion, driven not by thermal fluctuations but by internal stresses. A continuous time random walk framework is outlined in which the waiting times between displacements (jumps), generated by the dynamics of internal stresses, are described by the generalized $\mathrm{\Gamma }$ distribution. The associated generalized diffusion equation is then identified. The solution to this equation is obtained as an integral over an infinite series of Fox $H$ functions. The probability density function is identified as initially non-Gaussian, while at longer timescales Gaussianity is recovered. Likewise, the second moment displays a transient nature, shifting between subdiffusive and diffusive character. The potential application of this mathematical description to the quaking observed in several soft-matter systems is discussed briefly.

中文翻译：

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由内应力重新分布驱动的异常扩散

本文探讨了异常扩散的数学描述，异常扩散不是由热波动驱动，而是由内部应力驱动。概述了一个连续时间随机游走框架，其中由内应力动态产生的位移（跳跃）之间的等待时间由广义$\mathrm{\Gamma }$分布。然后确定相关联的广义扩散方程。该方程的解是作为对 Fox 的无穷级数的积分获得的$高$功能。概率密度函数最初被识别为非高斯函数，而在更长的时间尺度上，高斯函数被恢复。同样，第二时刻显示出一种瞬态性质，在亚扩散和扩散特征之间转换。简要讨论了这种数学描述在几个软物质系统中观察到的震动的潜在应用。