The JMAK (Johnson–Mehl–Avrami–Kolmogorov) equation is exponential equation inserted power-law behavior on the parameter, and is widely utilized to describe the relaxation process, the nucleation process, the deformation of materials and so on. Theoretically the power exponent is occasionally associated with the geometrical factor of the nucleus, which gives the integral power exponent. However, non-integral power exponents occasionally appear and they are sometimes considered as phenomenological in the experiment. On the other hand, the power exponent decides the distribution of step time when the equation is considered as the superposition of the step function. This work intends to extend the interpretation of the power exponent by the new method associating Shannon entropy of distribution of step time with the method of Lagrange multiplier in which cumulants or moments obtained from the distribution function are preserved. This method intends to decide the distribution of step time through the power exponent, in which certain statistical values are fixed. The Shannon entropy to which the second cumulant is introduced gives fractional power exponents that reveal the symmetrical distribution function that can be compared with the experimental results. Various power exponents in which another statistical value is fixed are discussed with physical interpretation. This work gives new insight into the JMAK function and the method of Shannon entropy in general.
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