豪斯道夫导数扩散模型的谱熵与累积谱熵

摘要

The cumulative spectral entropy in space is derived based on the spectral entropy of the Hausdorff derivative diffusion model for describing the spatial and temporal complexity of the anomalous diffusion process. The individual,the total spectral and the cumulative spectral entropies are investigated by varying the diffusion coefficient and the diffusion time. It is shown that the spectral and the cumulative spectral entropies increase with the decrease of the order of the time Hausdorff derivative α or the space Hausdorff derivative β and are characterized by a heavy tail. With the decrease of the diffusion time or the diffusion coefficient, the normal diffusion(α=1,β=1)sees a faster decay of the individual spectral entropy than the anomalous diffusion,and the corresponding spectral density becomes narrower. Thus,the spectral and the cumulative spectral entropies of the Hausdorff derivative diffusion model can reflect the heterogeneous structure of complex media and the uncertainty of the underlying diffusion process.%基于豪斯道夫导数扩散模型的空间谱熵推导描述反常扩散过程时空复杂程度的空间累积谱熵,并考察个体谱熵、谱熵、累积谱熵随时空豪斯道夫导数、扩散系数和扩散时间的变化情况.计算结果表明,谱熵与累积谱熵随时间豪斯道夫导数 α 或空间豪斯道夫导数 β 的减小而增大,且具有拖尾特征.此外,随着扩散时间 t 或扩散系数 Dα,β的减小,正常扩散对应的个体谱熵衰减的速率比反常扩散快,且对应的谱密度更窄.因此,豪斯道夫导数扩散模型的谱熵和累积谱熵均能够反映复杂介质的非均质特征和内部扩散过程的不确定性.