Error analysis of an acceleration corrected diffusion approximation of Langevin dynamics with background flow

TL;DR

The paper analyzes an acceleration-corrected diffusion approximation for Langevin dynamics with a background flow in the averaging regime, proving that the strong error is O(ε) and the weak error O(ε^2) relative to the true dynamics. It develops a Limiting Langevin Calculus framework and uses drift-correction analysis to bound the difference between the true trajectory and the diffusion-approximation, with rigorous strong and weak estimates. The authors also demonstrate, both analytically and numerically, that the corrected diffusion captures long-time, nonuniform stationary distributions that naive diffusion fails to reproduce, and provide comprehensive 1D and 2D numerical validations. The results advance understanding of diffusion-approximations for inertial particles in flows and offer robust tools for predicting population-level transport statistics and stationary states.

Abstract

We consider the problem of approximating the Langevin dynamics of inertial particles being transported by a background flow. In particular, we study an acceleration corrected advection-diffusion approximation to the Langevin dynamics, a popular approximation in the study of turbulent transport. We prove error estimates in the averaging regime in which the dimensionless relaxation timescale $\varepsilon$ is the small parameter. We show that for any finite time interval, the approximation error is of order $\mathcal{O}(\varepsilon)$ in the strong sense and $\mathcal{O}(\varepsilon^2)$ in the weak sense, whose optimality is checked against computational experiment. Furthermore, we present numerical evidence suggesting that this approximation also captures the long-time behavior of the Langevin dynamics.

Figures (5)

• Figure 1: The errors $\left| \int (u^\varepsilon(x,t)-g^\varepsilon(x,t)) \cos(kx) \, dx \right|$ of the diffusion approximation \ref{['eq:PDEApprox']} at time $T=1$ in $\log$ scale for Fourier modes $k=1,2,...,6$.
• Figure 2: The errors $\left| \int (\tilde{u}^\varepsilon(x,t)-g^\varepsilon(x,t)) \cos(kx) \, dx \right|$ of the naive approximation \ref{['eq:naive']} at time $T=1$ in $\log$ scale for Fourier modes $k=1,2,...,6$.
• Figure 3: Strong and weak errors in $\log$ scale via Monte-Carlo simulations between solution $X^\varepsilon_t$ of \ref{['eq:Langevin']} and $Z^\varepsilon_t$ of \ref{['eq:advection-diffusion']} at time $T=1$. 500,000 trajectories were simulated using SRIW1 method, tested against $\varphi=\cos x$ for the weak error.
• Figure 4: Long-time behaviors at $T=100$. Row 1 displays the contours of the densities from Monte-Carlo simulation and reconstructed from kernel density estimation: (A) is the density of $X^\varepsilon_T$ satisfying the Langevin equation \ref{['eq:Langevin']}, (B) is the density $Z^\varepsilon_T$ satisfying the approximation \ref{['eq:advection-diffusion']}. Row 2 displays the stationary solution $u^\varepsilon$ of PDE \ref{['eq:PDEApprox']}. (C) is the contour of $u^\varepsilon$. (D) is the heatmap of $u^\varepsilon$, together with the background flow.
• Figure 5: Long time behavior at $T = 100$ for different methods. (A) describes the density of \ref{['eq:advection-diffusion']}, (B) the density of \ref{['eq:Hasselman-simple']}, (C) the density of \ref{['eq:bakhtin-kifer']}, (D) the density of \ref{['eq:LWX']}.

Theorems & Definitions (40)

• Theorem 1.1
• Corollary 1.2
• Proposition 2.1
• Lemma 2.2
• proof
• proof : Proof of strong estimate in Proposition \ref{['prop:BrownianEst']}
• proof : Proof of weak estimate in Proposition \ref{['prop:BrownianEst']}
• Lemma 3.1
• Remark 3.2
• Lemma 3.3