利用Fox H-函数获得弛豫时间解析分布函数以分析阻抗谱的程序
Procedure for Obtaining the Analytical Distribution Function of Relaxation Times for the Analysis of Impedance Spectra using the Fox $H$-function
AI总结：
本文提出一种基于Fox H-函数和两次逆拉普拉斯变换的通用解析程序，用于从阻抗模型中获得弛豫时间分布函数，并在已知阻抗模型上验证了其有效性。
摘要：
通过将电化学阻抗谱数据拟合到等效电路模型来进行解释，一直是电化学中的标准分析方法。然而，将数据从频域反演为弛豫时间分布函数（DFRT）在阻抗数据分析中已获得相当大的关注，因为它无需先验知识即可揭示有关底层电化学过程的更详细信息。本文的重点是提供一个通用程序，以解析方式从阻抗模型中获得DFRT，假设以基本德拜弛豫模型为核。该程序首先用Fox H-函数表示阻抗函数，该函数具有许多有用的性质，特别是其拉普拉斯变换仍然是H-函数。由此，通过两次连续的逆拉普拉斯变换迭代获得DFRT。在此过程中，可以很容易地得到对阶跃激励的响应函数表达式。该程序在一些已知的阻抗模型上进行了测试和验证。
英文摘要：
The interpretation of electrochemical impedance spectroscopy data by fitting it to equivalent circuit models has been a standard method of analysis in electrochemistry. However, the inversion of the data from the frequency domain to a distribution function of relaxation times (DFRT) has gained considerable attention for impedance data analysis, as it can reveal more detailed information about the underlying electrochemical processes without requiring a priori knowledge. The focus of this paper is to provide a general procedure for obtaining analytically the DFRT from an impedance model, assuming an elemental Debye relaxation model as the kernel. The procedure consists of first representing the impedance function in terms of the Fox $H$-function, which possesses many useful properties particularly that its Laplace transform is again an $H$-function. From there the DFRT is obtained by two successive iterations of inverse Laplace transforms. In the passage, one can easily obtain an expression for the response function to a step excitation. The procedure is tested and verified on some known impedance models.
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