Overdamped Langevin过阻尼 Langevin
dq_t=-\nabla V(q_t)\,dt+\sqrt{2}\,dW_t
We use β = 1 and initialize the wrapped position from its invariant Gibbs law.取 β = 1，取模后的位置从不变 Gibbs 分布出发。
\mu(dq)=Z^{-1}e^{-V(q)}\,dq
Probability · Stochastic dynamics · Quantitative CLT概率论 · 随机动力学 · 定量中心极限定理
Berry–Esseen bounds for
periodic Langevin dynamics周期 Langevin 动力学的
Berry–Esseen 界
Selected personal research contributions · Manuscript in preparation个人研究贡献选集 · 稿件准备中
My work on quantitative Gaussian approximation, within a joint project with David P. Herzog, Owen Smith and Akshat Sachin Valse.本页展示我在定量高斯近似方面的研究。合作项目成员包括 David P. Herzog、Owen Smith 和 Akshat Sachin Valse。
How long until transport looks Gaussian?观察多久，位移才接近高斯分布？
A particle moves across a periodic landscape. Its long-time displacement has a Gaussian limit. We quantify the finite-time error—and how it depends on friction.粒子在周期势场中运动，其长时间位移具有高斯极限。我们研究有限时间内的近似误差，以及误差对摩擦系数的依赖。
01
Two models. One transport question.两种动力学，同一个输运问题。
Let V be a smooth potential on the periodic configuration space M. The wrapped position q stays in M; Q records its lift to Euclidean space.设 V 是周期构型空间 M 上的光滑势函数。取模后的位置 q 留在 M 中；Q 表示欧氏空间中的展开位置。
Underdamped Langevin欠阻尼 Langevin
\begin{aligned}dq_t&=p_t\,dt,\\dp_t&=-\nabla V(q_t)\,dt-\gamma p_t\,dt\\&\quad+\sqrt{2\gamma/\beta}\,dW_t.\end{aligned}
Here γ > 0 is friction and β > 0 is inverse temperature. Position and momentum start from equilibrium.其中 γ > 0 为摩擦系数，β > 0 为逆温度。位置和动量从平衡分布出发。
The observable研究的观测量
Y_T^F=\frac{F^\top(Q_T-Q_0)}{\sqrt T}
F is a fixed unit direction. We compare this scalar displacement with a centered Gaussian having its asymptotic variance. Kolmogorov distance measures the largest gap between their cumulative distribution functions.F 是固定单位方向。我们将这个标量位移与具有相应渐近方差的中心高斯分布比较。Kolmogorov 距离衡量两者累积分布函数的最大差异。
Notation and equilibrium assumptions记号与平衡初始条件
M=(L\mathbb T)^{dN},\qquad \mathbb T=\mathbb R/\mathbb Z
For the underdamped process, the invariant measure is proportional to exp[−β(V(q)+|p|²/2)] dq dp; it does not depend on γ. We write Y with superscript γ,F when identifying the friction is useful.欠阻尼过程的不变分布正比于 exp[−β(V(q)+|p|²/2)] dq dp，本身不依赖 γ。需要强调摩擦系数时，记作带上标 γ,F 的 Y。
\sigma_F^2=F^\top D F,\qquad \sigma_{\gamma,F}^2=F^\top D_\gamma Fd_K(\mu,\nu)=\sup_{x\in\mathbb R}|F_\mu(x)-F_\nu(x)|D and Dγ denote asymptotic covariance matrices.D 和 Dγ 表示渐近协方差矩阵。
02
Main results主要结果
Smooth periodic potentials · Equilibrium initial data · Fixed-time scalar projections · T ≥ 1光滑周期势函数 · 平衡初始分布 · 固定时刻的标量投影 · T ≥ 1
Overdamped Gaussian approximation过阻尼情形的高斯近似
d_K\!\left(\mathcal L(Y_T^F),\mathcal N(0,\sigma_F^2)\right)\le C_F T^{-1/2}
For every fixed unit direction F, the normalized displacement has a Berry–Esseen bound with a constant independent of observation time.对每个固定单位方向 F，归一化位移满足 Berry–Esseen 界，其中常数不依赖观察时间。
Underdamped Gaussian approximation欠阻尼情形的高斯近似
d_K\!\left(\mathcal L(Y_T^{\gamma,F}),\mathcal N(0,\sigma_{\gamma,F}^2)\right)\le C_{\gamma,F}T^{-1/2}
For each fixed γ > 0 and direction F, the same time rate holds, with positive asymptotic variance. The constant may depend on friction.对每个固定 γ > 0 和方向 F，得到相同的时间收敛阶，且渐近方差严格为正。常数可以依赖摩擦系数。
Proposition 11.1
A uniform estimate at low friction低摩擦下的统一估计
d_K\!\left(\mathcal L(Y_T^\gamma),\mathcal N(0,\sigma_\gamma^2)\right)\le \frac{C}{\sqrt{\gamma T}}
In one dimension, there exist γ₀ > 0 and C < ∞ such that this holds for every 0 < γ ≤ γ₀ and T ≥ 1. The constant C is independent of γ and T.在一维情形下，存在 γ₀ > 0 和有限常数 C，使此界对所有 0 < γ ≤ γ₀ 及 T ≥ 1 成立。C 不依赖 γ 和 T。
What changes when friction becomes small?摩擦系数变小时，发生了什么？
The estimate is organized by time measured in friction relaxation units:误差界取决于用摩擦弛豫时间衡量的观察时长：
u=\gamma T
Equivalently, the constant in Theorem 2 can be chosen with等价地，Theorem 2 中的常数可选为满足
C_\gamma\le C\gamma^{-1/2}
03
Numerical illustrations数值展示
Overdamped convergence to a Gaussian过阻尼情形的高斯收敛
The one-dimensional experiment in Section 7 of the manuscript: V(q) = cos(2πq), β = 1.论文第 7 节的一维实验：V(q) = cos(2πq)，β = 1。
−0.506
Reported log–log fit slope论文报告的双对数拟合斜率
9.87 × 10−3
Empirical Kolmogorov distance at T = 80T = 80 时的经验 Kolmogorov 距离
The observed log–log slope is −0.506, close to the T⁻¹ᐟ² time rate in Theorem 1.观测到的双对数斜率为 −0.506，接近 Theorem 1 中的 T⁻¹ᐟ² 时间收敛阶。
Simulation setup from the manuscript论文中的模拟设置
Metropolized Euler–Maruyama with Δt = 0.01 and 80,000 independent replicas. Positions are initialized uniformly on the torus, followed by 1,000 burn-in steps. The unwrapped displacement is reset after burn-in. Measurements are taken at nine geometrically spaced terminal times between T = 2 and T = 80.采用 Metropolized Euler–Maruyama，Δt = 0.01，80,000 条独立路径。位置先从环面上的均匀分布出发，经过 1,000 步预热后重置展开位移。在 T = 2 至 T = 80 之间的九个几何间隔终止时间记录结果。
Theorem 2 · Proposition 11.1
How friction changes the Gaussian approximation摩擦如何影响高斯近似
I simulated 80,000 equilibrium trajectories at each friction for V(q) = cos q, β = 1. The two views connect the time-dependent error with the friction dependence of its constant.对 V(q) = cos q、β = 1 的模型，我在每个摩擦系数下模拟了 80,000 条从平衡分布出发的路径。两张图分别展示误差随时间的变化，以及常数对摩擦的依赖。
Theorem 2
Time decay at fixed friction固定摩擦下的时间衰减
Proposition 11.1
The constant at low friction低摩擦下的常数
Across the four plotted frictions图中的四个摩擦系数
\gamma T\approx20\text{–}25
The first checkpoints after which d̂K ≤ 0.05 at every later sampled time.从这些检查点起，后续所有采样时刻均有 d̂K ≤ 0.05。
Small-friction reference小摩擦参考值
0.376 / 0.374
Energy-process simulation / extrapolation of the finite-friction constants.能量过程模拟 / 有限摩擦常数的外推。
Constant profiles & computation常数曲线与计算方法
From errors to constants从误差计算常数
\widehat C_\gamma=\max_{T\in\mathcal T_\gamma}\sqrt T\,\widehat d_K(\gamma,T)
\widehat c(\gamma)=\sqrt\gamma\,\widehat C_\gamma
Here 𝒯γ is the simulated time grid with T ≥ 1, refined around the peak. Error bars show basic bootstrap intervals from 100 trajectory resamples.𝒯γ 为满足 T ≥ 1 的模拟时间网格，在峰值附近加密。误差棒为对路径进行 100 次重采样得到的 basic bootstrap 区间。
Two calculations of the reference value参考值的两种计算
The averaged energy process gives 0.376, with bootstrap interval [0.370, 0.379], at rescaled time u ≈ 3.5. A weighted fit ĉ(γ) ≈ a + b√γ over γ ≤ 1/8 gives a = 0.374 and b = −0.220.平均化能量过程在重标定时间 u ≈ 3.5 处给出 0.376，bootstrap 区间为 [0.370, 0.379]。对 γ ≤ 1/8 的数据作加权拟合 ĉ(γ) ≈ a + b√γ，得到 a = 0.374、b = −0.220。
Simulation setup模拟设置
Finite-friction dynamics有限摩擦动力学
BAOAB, Δt = 0.05, one seed and 80,000 independent trajectories at each of eleven frictions γ = 4, 2, 1, 1/2, …, 1/256. Initial positions and momenta are sampled from the Gibbs distribution.采用 BAOAB，Δt = 0.05；γ = 4、2、1、1/2、…、1/256，共十一个摩擦系数，每个使用一个随机种子、80,000 条独立路径。初始位置和动量从 Gibbs 分布采样。
The Gaussian variance is estimated by extrapolating Var(QT − Q0)/T linearly in 1/T over the six largest terminal times, using all trajectories.使用全部路径，在最大的六个终止时间上，将 Var(QT − Q0)/T 关于 1/T 作线性外推，得到目标高斯分布的方差估计。
Averaged energy process平均化能量过程
80,000 trajectories on the graph of trapped and running energy orbits, with Euler step Δu = 0.001. The displacement accumulates on running orbits. The Gaussian reference has variance 2D₀, with D₀ = 0.1288501.在由受困轨道与运行轨道构成的能量图上模拟 80,000 条路径，Euler 步长 Δu = 0.001。位移在运行轨道上累积。参考高斯分布的方差为 2D₀，其中 D₀ = 0.1288501。
04
The proof, in three ideas三个证明思路
Separate the corrector分离校正项
Solve the Poisson equation. Write displacement as a martingale plus a bounded corrector. Apply a Markov-additive Berry–Esseen theorem and transfer the estimate.求解 Poisson 方程，将位移分解为鞅与有界校正项。应用 Markov 加性过程的 Berry–Esseen 定理，再传递误差估计。
Use the additive structure利用加性结构
Apply the Markov-additive Berry–Esseen theorem directly to integrated velocity, and combine it with moment and asymptotic-variance estimates.直接对速度的时间积分应用 Markov 加性过程的 Berry–Esseen 定理，并结合矩估计及渐近方差估计。
Sample at the relaxation scale按弛豫尺度采样
Take blocks of length comparable to γ⁻¹. Uniform mixing, fourth moments and a one-dimensional diffusion lower bound yield a uniform Berry–Esseen estimate.选取长度与 γ⁻¹ 同阶的时间块。利用统一混合、四阶矩及一维扩散下界，获得统一 Berry–Esseen 估计。
h\asymp\gamma^{-1}\quad\Longrightarrow\quad n\asymp\gamma T\quad\Longrightarrow\quad n^{-1/2}\asymp(\gamma T)^{-1/2}
05
Contributions & context贡献与研究背景
My contributions我的贡献
- Formulated the overdamped Berry–Esseen framework and proved its main results.建立过阻尼 Berry–Esseen 分析框架并证明主要结果。
- Proved the main underdamped theorem and supporting propositions through a direct Markov-additive argument.通过直接的 Markov 加性过程论证，证明欠阻尼主定理及相关命题。
- Established Proposition 11.1, tracking the low-friction dependence of the Berry–Esseen constant.证明 Proposition 11.1，追踪 Berry–Esseen 常数在低摩擦下的参数依赖。
- Simulated overdamped and underdamped transport, measured the friction-dependent constants, and compared them with the averaged energy dynamics.完成过阻尼与欠阻尼输运模拟，计算常数对摩擦的依赖，并与平均化能量动力学比较。
Supervised by David P. Herzog. The joint manuscript also includes contributions by Owen Smith and Akshat Sachin Valse.项目由 David P. Herzog 指导。合作稿件还包含 Owen Smith 与 Akshat Sachin Valse 的其他贡献。
Read at a glance一页了解项目
Download the print-ready research poster.下载可打印的研究海报。
36 × 24 inches · Landscape · English36 × 24 英寸 · 横向 · 英文
NSF-supported researchNSF 资助研究
DMS-2246491
Selected foundations主要理论文献
Ferré, Hervé & Ledoux (2012). Limit theorems for stationary Markov processes with L²-spectral gap.
Roussel & Stoltz (2018). Spectral methods for Langevin dynamics and associated error estimates.
Hairer & Pavliotis (2008). From ballistic to diffusive behavior in periodic potentials.
Theorem statements and the overdamped figure follow the October 2026 manuscript. The underdamped experiments and constant calculations were completed on 19 October 2026.定理陈述与过阻尼图依据 2026 年 10 月稿件。欠阻尼实验与常数计算完成于 2026 年 10 月 19 日。