| Abstract: |
| The derivation of macroscopic models from kinetic equations remains a cornerstone of statistical mechanics. For systems where the equilibrium density decays rapidly (e.g., Maxwellians), a diffusive scaling typically yields classical parabolic equations for the conserved quantities in the limit. However, it has been shown - primarily for single-conservation laws - that algebraic `fat-tailed` equilibria lead to anomalous transport characterized by fractional diffusion.
This talk extends these results to linear kinetic equations that conserve mass, momentum, and energy. We characterize the scaling limits based on the equilibrium decay rate: for sufficiently fast-decaying densities, we recover a system of classical diffusion equations. Conversely, for slower algebraic decay, we derive fractional diffusion equations for mass and energy, while the momentum equation becomes trivial.
The proof is based on spectral analysis and energy estimates. They are constructive and provide explicit convergence rates. |
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