Kosztołowicz Tadeusz
Institute of Physics, Jan Kochanowski University, Uniwersytecka 7, 25-406 Kielce, Poland.
Department of Radiological Informatics and Statistics, Medical University of Gdańsk, Tuwima 15, 80-210 Gdańsk, Poland.
Entropy (Basel). 2025 Jan 9;27(1):48. doi: 10.3390/e27010048.
Superdiffusion is usually defined as a random walk process of a molecule, in which the time evolution of the mean-squared displacement, σ2, of the molecule is a power function of time, σ2(t)∼t2/γ, with γ∈(1,2). An equation with a Riesz-type fractional derivative of the order γ with respect to a spatial variable (a fractional superdiffusion equation) is often used to describe superdiffusion. However, this equation leads to the formula σ2(t)=κt2/γ with κ=∞, which, in practice, makes it impossible to define the parameter γ. Moreover, due to the nonlocal nature of this derivative, it is generally not possible to impose boundary conditions at a thin partially permeable membrane. We show a model of superdiffusion based on an equation in which there is a fractional Caputo time derivative with respect to another function, ; the spatial derivative is of the second order. By choosing the function in an appropriate way, we obtain the -superdiffusion equation, in which Green's function (GF) in the long time limit approaches GF for the fractional superdiffusion equation. GF for the -superdiffusion equation generates σ2 with finite κ. In addition, the boundary conditions at a thin membrane can be given in a similar way as for normal diffusion or subdiffusion. As an example, the filtration process generated by a partially permeable membrane in a superdiffusive medium is considered.
超扩散通常被定义为分子的随机游走过程，其中分子的均方位移σ²的时间演化是时间的幂函数，即σ²(t)∼t²/γ，γ∈(1,2)。一个关于空间变量具有γ阶Riesz型分数阶导数的方程（分数阶超扩散方程）常被用于描述超扩散。然而，这个方程会导致σ²(t)=κt²/γ且κ=∞的公式，在实际中这使得无法定义参数γ。此外，由于这种导数的非局部性质，通常不可能在薄的部分渗透膜上施加边界条件。我们展示了一个基于这样一个方程的超扩散模型，该方程中存在关于另一个函数的分数阶Caputo时间导数，空间导数为二阶。通过以适当方式选择该函数，我们得到了-superdiffusion方程，其中长时间极限下的格林函数（GF）趋近于分数阶超扩散方程的GF。-superdiffusion方程的GF生成有限κ的σ²。此外，薄膜上的边界条件可以以与正常扩散或次扩散类似的方式给出。例如，考虑了部分渗透膜在超扩散介质中产生的过滤过程。
