Quantitative low-temperature spectral asymptotics for reversible diffusions in temperature-dependent domains

TL;DR

This work studies low-temperature spectral behavior of overdamped Langevin dynamics in domains whose boundaries depend on temperature, revealing how boundary proximity to energy saddles reshapes metastable timescales. By embedding the problem in the Witten Laplacian framework and constructing a harmonic approximation across critical points, the authors prove a leading-order harmonic spectrum limit and a modified Eyring–Kramers formula for the principal eigenvalue, explicitly incorporating boundary-distance parameters via a Gaussian-CDF factor. The analysis introduces geometric boundary-assumptions that localize boundary effects near critical points and develops a Laplace method for moving domains to handle boundary evolution. Practically, the results enable quantitative estimates of exit and decorrelation times for metastable diffusion and inform parameter-tuning in accelerated molecular dynamics methods like ParRep, with explicit prefactors depending on the boundary geometry and Hessian data at critical points.

Abstract

We derive novel low-temperature asymptotics for the spectrum of the infinitesimal generator of the overdamped Langevin dynamics. The novelty is that this operator is endowed with homogeneous Dirichlet conditions at the boundary of a domain which depends on the temperature. From the point of view of stochastic processes, this gives information on the long-time behavior of the diffusion conditioned on non-absorption at the boundary, in the so-called quasistationary regime. Our results provide precise estimates of the spectral gap and principal eigenvalue, extending the Eyring-Kramers formula. The phenomenology is richer than in the case of a fixed boundary and gives new insight into the sensitivity of the spectrum with respect to the shape of the domain near critical points of the energy function. Our work is motivated by--and is relevant to--the problem of finding optimal hyperparameters for accelerated molecular dynamics algorithms.

Figures (6)

• Figure 1: Local geometry of $\Omega_\beta$ in the neighborhood of a critical point $z_i$ close to the boundary and inside the domain $\Omega_\beta$. The relevant length scales are $\gamma(\beta)\ll \beta^{-\frac{1}{2}} \ll \delta(\beta)$.
• Figure 2: Depiction of a basin $\mathcal{A}(z_0)$ in dimension $d=2$. In solid lines, the set $\mathcal{S}(z_0)$ defined in \ref{['eq:vstar']}. The dotted line is the set $\partial\mathcal{A}(z_0)\setminus\mathcal{S}(z_0)$. The dashed line is the level set $\{V=V^\star\}$. There are eleven critical points including the minimum $z_0$, five index 1 saddle points $z_1,z_2,z_3,z_4$ and $z_5$. The point $z_1$ is a non-separating saddle point. The remaining points are index-2 saddle points (local maxima). Here $I_{\min}=\{2,3,4\}$, and $\mathcal{X}(z_0)=\{1\}$. Under \ref{['hyp:energy_well']}, it holds $\alpha^{(1)}=+\infty$. Note however that one could even change the sets $\mathcal{K}$ and $\Omega_\beta$ to have $z_6\not\in\mathcal{K}$ and hence not be considered as a critical point.
• Figure 3: Illustration of the hypothesis \ref{['hyp:energy_well']} around the basin depicted in Figure \ref{['fig:basin']}. In red, the boundary of a domain satisfying the geometric constraint \ref{['hyp:locally_flat']} (at a fixed value of $\beta$), but violating \ref{['hyp:energy_well']} is depicted. The boundary crosses the level set $\{V=V^\star\}$, and $z^*$ is therefore a low-energy generalized saddle point (see Section \ref{['subsec:genericity']} below) for this domain. The critical points $z_8$ and $z_{10}$ are assumed not to be in the englobing set $\mathcal{K}$. The points $z_0,z_1,z_2$ and $z_4$ are far from the boundary, while the others are close. Note that the orientation convention for $v^{(4)}_{1}$ is fixed by the geometry of $\mathcal{A}(z_0)$.
• Figure 4: Schematic representation of the extended domain $\Omega_\beta^+$ satisfying \ref{['eq:inclusion_perturbed_domain']}, depicted here in the vicinity of $z_i$, a critical point close to, but outside, the boundary of $\partial \Omega_\beta$.
• Figure 5: Construction of the quasimode \ref{['eq:global_quasimode']} in the neighborhood of the low-energy saddle point $z_i$, depicted in the adapted $y^{(i)}$ coordinates. Here, we depict the elements entering into the construction of $\psi_\beta^{+}$, in the case $\alpha^{(i)}<+\infty$. The shaded blue cone corresponds to the positive superlevel set of the quadratic form $Q_{-}$ from the proof of Proposition \ref{['prop:suff_condition_delta']}.

Theorems & Definitions (49)

• Proposition 1: LBLLP12
• Proposition 2: LBLLP12
• Remark 1
• Remark 2
• Remark 3
• Lemma 3
• proof
• Remark 4
• Theorem 4
• Theorem 5