Connecting complexity with spectral entropy using the Laplace transformed solution to the fractional diffusion equation

• Laplace transformed fractional diffusion equation is formulated via Fox’s HH-function.
• The derived solution is equivalent with the existing Lévy stable solution.
• The overall spectral entropy increases with the decreasing derivative orders αα and ββ.
• Spectral entropy is considered a measure to characterize the complexity of diffusion.

Analytical solutions to the fractional diffusion equation are often obtained by using Laplace and Fourier transforms, which conveniently encode the order of the time and the space derivatives (αα and ββ) as non-integer powers of the conjugate transform variables (ss, and kk) for the spectral and the spatial frequencies, respectively. This study presents a new solution to the fractional diffusion equation obtained using the Laplace transform and expressed as a Fox’s HH-function. This result clearly illustrates the kinetics of the underlying stochastic process in terms of the Laplace spectral frequency and entropy. The spectral entropy is numerically calculated by using the direct integration method and the adaptive Gauss–Kronrod quadrature algorithm. Here, the properties of spectral entropy are investigated for the cases of sub-diffusion and super-diffusion. We find that the overall spectral entropy decreases with the increasing αα and ββ, and that the normal or Gaussian case with α=1α=1 and β=2β=2, has the lowest spectral entropy (i.e., less information is needed to describe the state of a Gaussian process). In addition, as the neighborhood over which the entropy is calculated increases, the spectral entropy decreases, which implies a spatial averaging or coarse graining of the material properties. Consequently, the spectral entropy is shown to provide a new way to characterize the temporal correlation of anomalous diffusion. Future studies should be designed to examine changes of spectral entropy in physical, chemical and biological systems undergoing phase changes, chemical reactions and tissue regeneration.