Fractional kinetics equation from a Markovian system of interacting Bouchaud trap models
Fractional kinetics equation from a Markovian system of interacting Bouchaud trap models
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发表时间：
2023-02
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通讯作者：
Alberto Chiarini;Simone Floreani;F. Redig;Federico Sau
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作者：
Alberto Chiarini;Simone Floreani;F. Redig;Federico Sau
我们考虑了随机捕获环境中在$\mathbb Z^d$上演化的部分排斥过程.在$d\ge 2$维中，我们推导出分数动力学方程\开始{equation*}\frac{\partial^\beta\rho_t}{\partial t^\beta} = \Delta \rho_t \end{equation*}作为粒子系统的流体动力学极限。这里，$\frac{\partial^\beta}{\partial t^\beta}$，$\beta\in（0，1）$表示卡普托意义下的分数阶导数。因此，我们展示了一个马尔可夫相互作用的粒子系统，其经验密度场重新缩放到一个子扩散方程对应的非马尔可夫过程，分数动力学过程。相比之下，我们表明，当$d=1$，系统重新缩放到\开始{equation*} \frac{\partial \rho_t}{\partial t}= \mathcal L_\beta \rho_t\，\end{equation*}的解决方案，其中$\mathcal L_\beta$是称为FIN扩散的奇异准扩散的随机生成器。
We consider a partial exclusion process evolving on $\mathbb Z^d$ in a random trapping environment. In dimension $d\ge 2$, we derive the fractional kinetics equation \begin{equation*}\frac{\partial^\beta\rho_t}{\partial t^\beta} = \Delta \rho_t \end{equation*} as a hydrodynamic limit of the particle system. Here, $\frac{\partial^\beta}{\partial t^\beta}$, $\beta\in(0,1)$, denotes the fractional derivative in the Caputo sense. We thus exhibit a Markovian interacting particle system whose empirical density field rescales to a sub-diffusive equation corresponding to a non-Markovian process, the Fractional Kinetics process. In contrast, we show that, when $d=1$, the system rescales to the solution to \begin{equation*} \frac{\partial \rho_t}{\partial t}= \mathcal L_\beta \rho_t\ , \end{equation*} where $\mathcal L_\beta$ is the random generator of the singular quasi-diffusion known as FIN diffusion.