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The stretched exponential function, $\exp[-(t/\tau_{K})^{\beta}]$, describesvarious relaxation processes while it has been suggested that the powerexponent, $\beta$ is derived from the non-uniformity of the process. In thispaper, we attempted to estimate this non-uniformity by introducing Shannonentropy. Shannon entropy evaluates the average information contents of thedistribution function, which reflects statistical homogeneity. We investigatedthe relaxation process of shear induced multilamellar system, which isdescribed with the stretched exponential function. Three types of shear(constant, square, sine) at different frequencies are attempted in order todetermine their effects on the relaxation process. We found that the Shannonentropy to which the first moment was introduced is maximized at $\beta~=~1$ :a single exponential. The Shannon entropy of sine shear experiments exhibitedthe frequency dependence. Thus it is interpreted that the increase of theintensity of shearing and the thermodynamic entropy are reflected on theShannon entropy. We discussed the meaning of the maximum Shannon entropy interms of various points of view, it was found that it corresponds to thediffusion process free from the restriction such as geometrical constraints.The constraints and the non-uniformity of process were successfully estimatedby Shannon entropy. This study gives a new insight of entropy in general.

Shannon entropy analysis of the stretched exponential process : application to various shear induced multilamellar vesicles system