- Fractional coalescent
An approach to the coalescent, the fractional coalescent......
An approach to the coalescent, the fractional coalescent (f-coalescent), is introduced. The derivation is based on the discrete-time Cannings population model in which the variance of the number of offspring depends on the parameter α. This additional parameter α affects the variability of the patterns of the waiting times; values of α<1 lead to an increase of short time intervals, but occasionally allow for very long time intervals. When α=1, the f-coalescent and the Kingman’s n-coalescent are equivalent. The distribution of the time to the most recent common ancestor and the probability that n genes descend from m ancestral genes in a time interval of length T for the f-coalescent are derived. The f-coalescent has been implemented in the population genetic model inference software MIGRATE. Simulation studies suggest that it is possible to accurately estimate α values from data that were generated with known α values and that the f-coalescent can detect potential environmental heterogeneity within a population. Bayes factor comparisons of simulated data with α<1 and real data (H1N1 influenza and malaria parasites) showed an improved model fit of the f-coalescent over the n-coalescent. The development of the f-coalescent and its inclusion into the inference program MIGRATE facilitates testing for deviations from the n-coalescent.
- A physical interpretation of fractional viscoelasticity based on the fractal structure of media: Theory and experimental validation
S.Mashayekhi,M.Yousuff Hussaini and W.Oates
In this work, a physical connection between the fractional time......
In this work, a physical connection between the fractional time derivative and fractal geometry of fractal media is developed and applied to viscoelasticity and thermal diffusion in elastomers. Integral to this formulation is the application of both the fractal dimension and the spectral dimension which characterizes diffusion in fractal media. The methodology extends the generalized molecular theory of Rouse and Zimm where generalized Gaussian structures (GGSs) replace the Rouse matrix with the generalized Gaussian Rouse matrix (GRM). Importantly, the Zimm model is extended to fractal media where the new relaxation formulation contains internal state variables that naturally depend on the fractional time derivative of deformation. Through the use of thermodynamic laws in fractal media, we derive the linear fractional model of viscoelasticity based on both spectral and fractal dimensions. This derivation shows how the order of the fractional derivative in the linear fractional model of viscoelasticity is a rate dependent material property that is strongly correlated with fractal and spectral dimensions in fractal media. To validate the correlation between fractional rates and fractal material structure, we measure the viscoelasticity and thermal diffusion of two different dielectric elastomers: Very High Bond (VHB) 4910 and VHB 4949. Using Bayesian uncertainty quantification (UQ) based on uniaxial stress–strain measurements, the fractional order of the derivative in the linear fractional model of viscoelasticity is quantified. Two dimensional fractal dimensions are also independently quantified using the box counting method. Lastly, the diffusion equation in fractal media is inferred from experiments using Bayesian UQ to quantify the spectral dimension by heating the polymer locally with a laser beam and quantifying thermal diffusion. Comparing theory to experiments, a strong correlation is found between the viscoelastic fractional order obtained from stress–strain measurements in comparisons with independent predictions of fractional viscoelasticity based on the fractal structure and fractional thermal diffusion rates.