In this talk, we study fast diffusion equations (FDEs) in the context of interacting particle systems (IPS). The term fast diffusion refers to the fact that the diffusion coefficient diverges as the density approaches zero. These equations have been extensively studied in the literature and arise in a wide range of physical applications. For instance, they model diffusion in plasmas, appear in the study of cellular automata and interacting particle systems exhibiting self-organized criticality, and describe the evolution of plane curves shrinking along the normal direction at a curvature-dependent speed.

From the perspective of interacting particle systems, the first part of the talk is devoted to deriving an FDE as the scaling limit of a sequence of zero-range processes with symmetric unit rates. To capture the fast diffusion behavior at the microscopic level, we introduce an appropriate rescaling of models featuring a typically large number of particles per site. In the second part, we introduce a family of zero-range processes aimed at establishing a connection between the results of Landim (1996), Morris (2006), and Nagahata (2010). Certain processes within this family are naturally associated with fast diffusion equations, and our main goal is to determine the order of the relaxation time, a key ingredient in the derivation of scaling limits. Starting from a heuristic argument that estimates the relaxation time of a general zero-range process in terms of its partition function, we identify a parametric family of partition functions arising as solutions of a specific ordinary differential equation. By analyzing the asymptotic behavior of the coefficients in their power series expansions, we derive the corresponding family of rate functions. Finally, we present numerical evidence—obtained via deterministic iterative methods and Monte Carlo simulations—supporting the predicted order of the relaxation time for these processes.

This is joint work with Milton Jara (IMPA) and Freddy Hernández (UFF / Universidad Nacional de Colombia).

Here we introduce basic concepts, various models (SIP, SEP, independent random walkers) and how they are linked to each other via the Lie algebraic formalism.

From the Lie algebraic formalism we infer that interacting particle systems with dualities come in "families" characterized by an underlying Lie algebra.

These are SU(2) for SEP, SU(1,1) for SIP, and the Heisenberg algebra for independent particles.

References

1. Giardina, C., & Redig, F. (2026). Duality for Markov processes: a Lie algebraic approach. Springer Nature.
2. Van Ginkel, B., & Redig, F. (2020). Hydrodynamic Limit of the Symmetric Exclusion Process on a Compact Riemannian Manifold: B. van Ginkel et al. Journal of Statistical Physics, 178(1), 75-116.
3. Junné, J., Redig, F., & Versendaal, R. (2024). Hydrodynamic limit of the symmetric exclusion process on complete Riemannian manifolds and principal bundles. arXiv:2410.20167.
4. Giardinà, C., Redig, F., & van Tol, B. (2024). Intertwining and propagation of mixtures for generalized KMP models and harmonic models. arXiv:2406.01160.
5. Schütz, G., & Sandow, S. (1994). Non-Abelian symmetries of stochastic processes: Derivation of correlation functions for random-vertex models and disordered-interacting-particle systems. Physical Review E, 49(4), 2726.
6. Giardina, C., Kurchan, J., Redig, F., & Vafayi, K. (2009). Duality and hidden symmetries in interacting particle systems. Journal of Statistical Physics, 135(1), 25-55.
7. Frassek, R., & Giardinà, C. (2022). Exact solution of an integrable non-equilibrium particle system. Journal of Mathematical Physics, 63(10).

Here we use duality to characterize the ergodic invariant measures, and use duality to also look at the stationary state of systems driven by reservoirs at the boundary.

Special attention is given to the harmonic model and propagation of mixed product states.

Here we use duality to characterize hydrodynamic limits and fluctuation fields.

Special attention is given to the hydrodynamic limit of SEP in a geometric setting, i.e., on graphs that approximate a Riemannian manifold.