Critically-Damped Higher-Order Langevin Dynamics

• Critically-damped higher-order Langevin dynamics are an extension of stochastic differential equations that incorporate auxiliary variables for optimal non-oscillatory convergence.
• The framework utilizes eigenvalue tuning and matrix exponentiation to simplify system analysis, ensuring rapid mixing and smooth sample trajectories.
• This approach underpins advances in generative modeling, uncertainty quantification, and molecular simulation by significantly reducing computational time and error.

Critically-damped higher-order Langevin dynamics generalize the concept of critical damping, central to systems analysis and control theory, to stochastic differential equations (SDEs) of arbitrary order. The approach leverages auxiliary variables—such as velocity, acceleration, or even higher-order derivatives—to construct multi-dimensional diffusion processes whose parameters are tuned to minimize the rate-limiting eigenvalue of the system’s forward operator. The resulting dynamics exhibit optimal convergence properties, rapid mixing, and smooth sample trajectories, which are essential in state-of-the-art generative modeling, uncertainty quantification, and molecular simulation.

1. Mathematical Framework of Critically-Damped Higher-Order Langevin Dynamics

The extension to arbitrary order n introduces auxiliary state variables to capture not only position but higher derivatives, leading to a general Itô SDE system:

$$d\mathbf{x}_t = \mathcal{F} \mathbf{x}_t dt + \mathcal{G} d\mathbf{w}_t$$

where $\mathbf{x}_t \in \mathbb{R}^{nh}$ concatenates the primary variable and $n-1$ auxiliary variables in $\mathbb{R}^h$, and $\mathcal{F}$, $\mathcal{G}$ are block matrices:

• The drift matrix $F$ is constructed as

$$F = \sum_{i=1}^{n-1} \gamma_i (E_{i,i+1} - E_{i+1,i}) - \xi E_{n,n}$$

where $E_{i,j}$ are elementary matrices, $\gamma_i$ are system parameters, and $\xi$ is a damping parameter.

• The diffusion matrix $G=\sqrt{2\xi L^{-1}} E_{n,n}$ injects noise only in the highest auxiliary variable.

The evolution of the forward mean and covariance is then

$$\frac{d\mu_t}{dt} = \mathcal{F}\mu_t, \quad \frac{d\Sigma_t}{dt} = \mathcal{F}\Sigma_t + (\mathcal{F}\Sigma_t)^\top + \mathcal{G}\mathcal{G}^\top$$

with closed-form solutions: $$\mu_t = \exp(\mathcal{F}t)x_0, \quad \Sigma_t = L^{-1}I + \exp(\mathcal{F}t)(\Sigma_0 - L^{-1}I)\exp(\mathcal{F}t)^\top$$ The above framework arises naturally in higher-order Denoising Diffusion Probabilistic Models (DDPMs) and Markov Chain Monte Carlo (MCMC) samplers (Sterling et al., 26 Jun 2025).

2. Critical Damping Principle and Parameter Selection

Critical damping in the context of SDEs is defined via the eigenstructure of the drift matrix $F$. The critical damping condition is achieved when $F$ has a single real, degenerate eigenvalue $\lambda^*$ (geometric multiplicity one), leading to the fastest non-oscillatory convergence:

• For order-$n$ system, parameters are set by

$$\gamma_{n-i} = |\lambda^*|\sqrt{\frac{n^2-i^2}{4i^2-1}} ~~~~ (i=1,\ldots,n-1),\quad \xi = n|\lambda^*|$$

Fixing $\gamma_1=1$, the unique eigenvalue is $\lambda^* = -\sqrt{2n-3}$.

• The critical damping ensures every eigenvalue of $F$ equals $-\xi/n$, meaning all transient modes decay at the same maximal rate.
• Matrix exponentiation simplifies: Instead of requiring a spectral decomposition, for repeated eigenvalues,

$$\exp(F t) = \exp(\lambda^* t) \sum_{k=0}^{n-1} \frac{(F-\lambda^* I)^k t^k}{k!}$$

This not only facilitates efficient simulation but also improves stability and reversibility in consistent SDE samplers (Sterling et al., 12 Sep 2024, Sterling et al., 26 Jun 2025).

3. Forward and Reverse Processes in Diffusion Models

Critically-damped higher-order Langevin dynamics are particularly effective in denoising diffusion models used for generative modeling. The forward process lifts the original variable $q_0$ into an $n$-dimensional extended state $\mathbf{x}_0 = (q_0, p_0, s_0, ...) ^T$, where auxiliary variables are typically initialized independently from $\mathcal{N}(0, \beta L^{-1} I)$:

• The forward SDE:

$$d\mathbf{x}_t = \mathcal{F} \mathbf{x}_t dt + \mathcal{G} d\omega$$

with the mean and covariance propagated as above.

• The reverse process (score-guided backward SDE) leverages analytical mean/covariance, and critical damping produces smooth, non-oscillatory paths, facilitating high-quality generative models with reduced computational cost (Dockhorn et al., 2021, Shi et al., 19 Apr 2024, Sterling et al., 12 Sep 2024, Sterling et al., 26 Jun 2025).

4. Accelerated Mixing, Sample Quality, and Computational Advantages

Critical damping optimally balances inertia and dissipation, yielding several practical benefits:

• Accelerated Convergence: All modes decay exponentially at the fastest non-oscillatory rate permitted by the model structure, reducing mixing time by up to two orders of magnitude compared to lower-order or non-critically damped schemes (Sterling et al., 26 Jun 2025, Shi et al., 19 Apr 2024).
• Smooth Trajectories: Noise is injected in higher auxiliary variables (e.g., acceleration), which "filters" stochasticity before it reaches the data variable, leading to smoother, more stable trajectories that reduce discretization error (Dockhorn et al., 2021, Shi et al., 19 Apr 2024).
• Simple Analytical Forms: Parameters set by the critical damping conditions enable more efficient evaluation of mean/covariance updates and matrix exponentials, improving simulation efficiency and stability (Sterling et al., 26 Jun 2025, Sterling et al., 12 Sep 2024).
• Empirical Impact: Applications include denoising diffusion models in image and speech synthesis with improved Fréchet Inception Distance (FID) and log-likelihood; e.g., FID values as low as 1.85 were obtained on CIFAR-10 (Shi et al., 19 Apr 2024, Sterling et al., 12 Sep 2024, Sterling et al., 26 Jun 2025).

5. Adaptive Algorithms and Measure Preservation

In numerical integration, critical damping often renders a system locally stiff, especially in higher dimensions or near sharp features in the potential landscape. Recent methods adapt the time step via a "monitor function" $g(x)$ to preserve the invariant measure, even in the presence of highly state-dependent or rapidly changing drift:

• The invariant measure-preserving transformed SDE for the overdamped case is:

$$dx = -g(x) \nabla V(x) dt + \frac{1}{\beta} \nabla g(x) dt + \sqrt{\frac{2}{\beta}g(x)} dW$$

• Higher-order/underdamped dynamics receive analogous corrections to ensure that the stationary Gibbs measure is preserved under adaptive step size and critical damping (Leroy et al., 18 Mar 2024).

This approach allows for larger steps in "easy" regions and smaller steps in stiff or rapidly changing regions, which is particularly critical for higher-order, critically-damped systems.

6. Applications: Generative Modeling, Inverse Problems, and Privacy

Generative Modeling: Critically-damped higher-order Langevin dynamics underpin recent advances in diffusion probabilistic models for image, speech, and data generation. They improve sample quality, speed up training and sampling, and reduce the complexity of the score-matching task (Dockhorn et al., 2021, Shi et al., 19 Apr 2024, Sterling et al., 12 Sep 2024, Sterling et al., 26 Jun 2025).

Inverse Problems: In applications such as MIMO symbol detection, channel estimation, and image reconstruction, higher-order (second- and third-order) critically-damped Langevin samplers with preconditioning and annealing outperform overdamped or underdamped schemes, both in accuracy and convergence speed (Zilberstein et al., 2023).

Privacy Defense: Critically-damped higher-order dynamics can be used to defend diffusion models against membership inference attacks. By augmenting the forward diffusion SDE with auxiliary variables and tuning their variance, these models introduce external randomness that effectively corrupts sensitive input data early in the diffusion process. Theoretical analysis shows that the privacy loss (Rényi Divergence) is maximized at $t=0$ and decreases monotonically, and experiments show reduced AUROC for attacks—even at the limit of random guessing—while maintaining or improving output quality (Sterling et al., 17 Sep 2025).

7. Theoretical and Practical Implications

The adoption of critical damping in higher-order Langevin dynamics is mathematically optimal in the sense that it maximizes the real parts of the eigenvalues of the drift operator, thereby minimizing the slowest decay rate of transients. The closed-form expressions for the involved matrix exponentials and system parameters streamline both theoretical analysis and algorithmic implementation.

These frameworks provide a rigorous and extensible foundation applicable beyond classical diffusion; they generalize seamlessly to arbitrary order, facilitate exact computation of forward transition statistics, and adapt naturally to the requirements of modern stochastic modeling in high-dimensional spaces.

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System Order	Key Parameters at Critical Damping	Matrix Exponential Simplification	Main Application Domains
2	$\gamma = 2\omega$	Single exponential	Imaging, SDE samplers
3	$\gamma = 2\sqrt{2}$, $\xi = 3\sqrt{3}$	Polynomial times single exponential	Accelerated DDPMs, MCMC
n	$\gamma_{n-i}, \xi$ as above	Degree $n-1$ polynomial times single exp.	Arbitrary-order DDPMs

Critical damping in higher-order Langevin dynamics is thus an effective and theoretically grounded strategy for enhanced convergence, smoothness, and robustness in applications ranging from molecular simulation, uncertainty quantification, generative modeling, to privacy-preserving data generation (Sterling et al., 26 Jun 2025, Sterling et al., 12 Sep 2024, Sterling et al., 17 Sep 2025).