Speaker
Description
A Monte Carlo algorithm is proposed for solving subdiffusion equations with a time-fractional derivative whose order depends on time or coordinate. The algorithm is based on the physical interpretation of the equation within the multiple-trapping model with an exponential energy distribution of localized states. The width of distribution $\varepsilon_0(x,t)$ may depend on coordinate or time. The corresponding subdiffusion equation contains a pseudo-differential operator of order $\alpha(x,t) = kT/\varepsilon_0(x,t)\in(0,1]$, consistent with the well-known Lorenzo-Hartley definition. The algorithm generates trajectories where the particle undergoes biased Brownian motion, then is trapped at an energy level $E$ drawn from the exponential distribution $\rho(\varepsilon; x,t) = \varepsilon_0^{-1}(x,t) e^{-\varepsilon/\varepsilon_0(x,t)}$, and the localization time is $\tau_l = -\nu_0^{-1} e^{E/kT}\ln U_2$ (where $U_2$ is a random variable uniformly distributed on $(0,1]$). An ensemble of $N$ independent trajectories is simulated, followed by the computation of statistical moments of the desired concentration. To accelerate sampling, parallelization over trajectories with independent streams of pseudorandom numbers is employed, making the algorithm highly scalable. The method enables the calculation of mean squared displacement, spatial distributions, and transient currents for variable-order subdiffusive regimes. For a constant subdiffusion order, the algorithm yields numerical solutions that coincide with known analytical solutions. Computational experiments convincingly confirm the correct reproduction of transitions from normal diffusion to subdiffusion and back.
