A collision-oriented interacting particle system for Landau-type equations and the molecular chaos

TL;DR

This paper introduces a collision-oriented particle system for approximating Landau-type equations, emphasizing physical realism and computational efficiency, with proven convergence and entropy-based analysis.

Contribution

It presents a novel particle system derived from grazing collision regimes, with a unique batch size of p=2, and demonstrates improved computational cost and convergence analysis.

Findings

The proposed system converges to Landau-type equations under regular interaction kernels.

It reduces computational complexity to O(N) per time step.

Gradient estimates are used to analyze convergence and stability.

Abstract

We propose a collision-oriented particle system to approximate a class of Landau-type equations. This particle system is formally derived from a particle system with random collisions in the grazing regime, and happens to be a special random batch system with random interaction in the diffusion coefficient. The difference from usual random batch systems with random interaction in the drift is that the batch size has to be $p=2$. We then analyze the convergence rate of the proposed particle system to the Landau-type equations using the tool of relative entropy, assuming that the interaction kernels are regular enough. A key aspect of our approach is the gradient estimates of logarithmic densities, applied to both the Landau-type equations and the particle systems. Compared to existing particle systems for the approximation of Landau-type equations, our proposed system not only offers a more intrinsic reflection of the underlying physics but also reduces the computational cost to $O(N)$ per time step when implemented numerically.