Linear Landau equation as a limit of a tagged particle in mean field interaction with a free gas

TL;DR

This work derives a diffusion limit for a tagged particle in a mean-field interacting gas at equilibrium in dimensions $d\ge 4$, showing convergence to a process $\mathcal{V}_\tau$ solving $d\mathcal{V}_\tau=2\Lambda(\mathcal{V}_\tau)\,d\tau+\sqrt{2}\Sigma(\mathcal{V}_\tau)dB_\tau$ whose associated Fokker-Planck equation is the linear Landau equation with $f_0=g_0\gamma$. The analysis hinges on a pathwise, martingale-problem approach combined with a careful control of correlations, long-time stability, and rare recollisions, along with a bootstrap averaging that extends the timescale from $N^{1/4}$ to $N^{1/3}$. Central to the argument are the explicit macroscopic coefficients $\Lambda$ and $D$, expressed via integrals and Fourier representations, and the demonstration that $\Lambda(V)=-D(V)V=\nabla\cdot D(V)$. The framework relies on the grand canonical ensemble to supply stationarity and uses a hierarchy of good/better sets to bound fluctuations and interactions, ultimately establishing tightness and uniqueness in the limiting martingale problem. This work links microscopic mean-field dynamics to Landau-type diffusion, illuminating how collective fluctuations in dense gases yield a linear Landau diffusion regime with potential insights for Lenard–Balescu-type limits.

Abstract

We consider a tagged particle in mean field interaction with a free gas of density N at equilibrium. In dimensions $d\geq4$, we prove the convergence of its trajectory, as N goes to infinity, to the one of a diffusion process associated with the linear Landau equation. The proof of the convergence of the martingale problem relies on two key ingredients: long time stability results of the microscopic dynamics, and controls on the probability of particle recollisions.

Figures (6)

• Figure 1: Illustration of the process $(Z_t)_t=(X_t,V_t)_t$ defined in \ref{['eq:def_tagged']}. The background particles move in a straight line, until they meet the tagged particle. The latter interacts with the background inside an interaction ball of radius $R$.
• Figure 2: Appearance of a friction. This figure depicts two symmetric trajectories of background particles (in black) deflected by a repulsive interaction with the tagged particle (in red). The main contribution of the force (solid black arrow, of order $N^{-1}$) created by a background particle cancels out on average. This leads to a random force of mean 0 and variance $1/N$ acting on the tagged particle. However, the deviation of order $N^{-1}$ of the background particles shifts slightly this force (solid red arrow) and ultimately induces a friction of order $N^{-2}$. Since during a time interval $[0,t]$ the tagged particle meets $\sim Nt$ background particles (each yielding this tiny force of strength $N^{-2}$), for $t\simeq N$, a friction force of order $1$ appears.
• Figure 3: Sources of time correlation. A background particle interacting for a long time, and a background particle recolliding. Both the interaction time and the first possible recollision time are controlled by the relative velocity.
• Figure 4: Proof of the maximum interaction time and minimum recollision time. Assuming $x^1_0\in\mathcal{B}(X_0,R)$, we compare $X_t$ to $X_0+tV_0$ and $x^1_t$ to $x^1_0+tv^1_0$. In the case of $\mathcal{G}_N(\delta)$, the possible values for $X_t$ must be in between the two red dashed lines, which represent the bound on the error. We are interested in the time at which, despite the error, the comparison with straight lines ensures that the background particle exits the interaction radius, and the time at which the error becomes so big that the tagged particle may re-interact with particle $1$. For an initial difference of velocities which is too small, the known error does not allow us to ensure that the difference of positions becomes larger than $2R$.
• Figure 5: Illustration of the proof of Proposition \ref{['prop:fin_bootstrap']} for $d=4$, only two iterations are needed.

Theorems & Definitions (78)

• Lemma 1.1
• Lemma 1.2
• Theorem 1
• Corollary 1.1
• Proposition 2.1
• Remark 2.1
• proof : Proof of Proposition \ref{['prop:martingale_formulation']}
• Definition 1: Interaction time
• Lemma 2.1
• proof : Proof of Lemma \ref{['lem:abstract_tout_va_bien']}