The New Method Using Shannon Entropy to Decide the Power Exponents on JMAK Equation

Hirokazu Maruoka

doi:10.3390/ecea-5-06660

Abstract

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.

Highlights

IntroductionThe stretched exponential function is described as follows, " β #
The stretched exponential function is described as follows, " β #t f (t, K, β) = exp − K (1)It is widely applied to describe relaxation processes [1,2,3,4,5], kinetics of crystallization [6,7,8,9], deformations of materials [10,11] and so on
The point that it is different from general maximum entropy estimation [32] is that the form of the equation is already fixed in Equation (1), the JMAK equation

Summary

Introduction

The stretched exponential function is described as follows, " β #. The name “stretched exponentials” corresponds to the case for β < 1, while the opposite case of β > 1 is called the “compressed exponential function”. The latter case corresponds to the JMAK (Johnson–Mehl–Avrami–Kolmogorov) equation [6,7,8,9], and β is sometimes called the Avrami constant. In the context of the JMAK equation, theoretically it is occasionally associated with the geometrical factor of the nucleus. The Avrami constant β originates from the dimensionality in which nucleation occurs D, which has the following relation, β = D + 1.

Methods

Discussion

Conclusion