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Physical Interpretation
Collier [91] applied Levy-flight to velocity space diffusion and generated Cauchy tails as expected.
Non-Markovian processes have "memory", or correlations in the time-domain. E.g., the lamppost is on a hill. That is, collisions have non-local information=> fractional calculus.
Time-fractional & Space-fractional diffusion equations are equivalent (if there is a velocity somewhere).
Non-local interactions, and/or non-Markovian interactions both produce fractional diffusion.
Many physical systems exhibit super/sub diffusion with fat tails.
Non-adiabatic systems need not conserve energy. E.g. if we maximize entropy holding log(E) constant => power law tails!