Fractional Advection-Diffusion-Asymmetry Equation

Area of Science:

• Physics
• Applied Mathematics
• Geophysics

Background:

• Fractional kinetic equations model anomalous relaxation and diffusion.
• Existing models fail to describe certain transport phenomena in disordered systems, particularly contaminant spreading.

Purpose of the Study:

• To propose and investigate a new fractional advection-diffusion equation.
• To describe biased transport packets in disordered systems.

Main Methods:

• Developing a fractional advection-diffusion equation based on continuous time random walks.
• Incorporating a finite mean waiting time and diverging variance in the random walk model.
• Analyzing the emergence of a symmetry-breaking term.

Main Results:

• The proposed equation successfully models biased spreading packets.
• A novel symmetry-breaking term is identified, essential for transport in disordered systems.
• The model accounts for long trapping times, explaining fractional space derivatives.

Conclusions:

• The new fractional advection-diffusion equation provides a more comprehensive description of anomalous transport.
• This work bridges a gap in fractional calculus literature by addressing systems with finite mean waiting times and diverging variances.
• The findings have implications for understanding contaminant transport in geological formations and other disordered media.