Random death process for the regularization of subdiffusive fractional equations.
Random death process for the regularization of subdiffusive fractional equations.
复制标题
次扩散分数方程正则化的随机死亡过程。
DOI：
10.1103/physreve.87.052139
复制
发表时间：
2012
期刊：
影响因子：
--
通讯作者：
Steven E Falconer
中科院分区：
文献类型：
--
作者：
S. Fedotov;Steven E Falconer
对于平稳分布，用常反常指数分数次方程来描述复杂介质中的次扩散输运是不稳健的。Gibbs-Boltzmann分布被反常指数的微小空间扰动从根本上改变[S.Fedotov和S.Falconer，Phys.修订本E 85,031132(2012年)]。为了纠正这个问题，我们提出在随机游走方案中加入随机死亡过程，这对于包括形态梯度形成在内的生物应用来说是非常自然的。由此，我们得到了修正的分数阶主方程，并通过解析和蒙特卡罗模拟分析了它的渐近行为。我们证明了该方程对反常指数的空间变化是结构稳定的。我们发现粒子的定常通量具有马氏形式，其速率函数依赖于反常速率函数、死亡率和反常指数。此外，在连续极限下，我们得到了一个平流扩散方程，其中平流系数和扩散系数同时取决于死亡率和反常指数。
The description of subdiffusive transport in complex media by fractional equations with a constant anomalous exponent is not robust where the stationary distribution is concerned. The Gibbs-Boltzmann distribution is radically changed by even small spatial perturbations to the anomalous exponent [S. Fedotov and S. Falconer, Phys. Rev. E 85, 031132 (2012)]. To rectify this problem we propose the inclusion of the random death process in the random walk scheme, which is quite natural for biological applications including morphogen gradient formation. From this, we arrive at the modified fractional master equation and analyze its asymptotic behavior, both analytically and by Monte Carlo simulation. We show that this equation is structurally stable against spatial variations of the anomalous exponent. We find that the stationary flux of the particles has a Markovian form with rate functions depending on the anomalous rate functions, the death rate, and the anomalous exponent. Additionally, in the continuous limit we arrive at an advection-diffusion equation where advection and diffusion coefficients depend on both the death rate and anomalous exponent.