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Separation of time scales in weakly interacting diffusions

Zachary P. Adams, Maximilian Engel, Rishabh S. Gvalani

TL;DR

The paper analyzes metastable behavior of $N$ weakly interacting Brownian particles on $\mathbb{R}^d$ with a localised attractive potential. By projecting out the centre-of-mass and studying the killed, projected diffusion through its generator, the authors connect the droplet state to the quasi-stationary distribution $q_N$ and use spectral analysis to quantify time-scale separation. They establish that the convergence to the droplet (QSD) occurs at rate $O(1)$ in inverse temperature $\beta$, while leakage from the droplet occurs at rate $O(e^{-\beta})$, and that the droplet is localized on length scale $O(\beta^{-1/2})$. Using a Schrödinger/Witten Laplacian framework and Simon’s norm-resolvent techniques, they obtain low-temperature asymptotics with $\lambda_{1,\beta,N}$ decaying exponentially and $\lambda_{2,\beta,N} \to w''(0)$, together with a precise description of multiscale convergence to the QSD and droplet dynamics. The results provide a rigorous microscopic counterpart to the aggregation-diffusion metastability questions and yield a quantitative, multiscale picture of droplet formation and decay.

Abstract

We study metastable behaviour in systems of weakly interacting Brownian particles with localised, attractive potentials which are smooth and globally bounded. In this particular setting, numerical evidence suggests that the particles converge on a short time scale to a "droplet state" which is $metastable$, i.e. persists on a much longer time scale than the time scale of convergence, before eventually diffusing to $0$. In this article, we provide rigorous evidence and a quantitative characterisation of this separation of time scales. Working at the level of the empirical measure, we show that (after quotienting out the motion of the centre of mass) the rate of convergence to the quasi-stationary distribution, which corresponds with the droplet state, is $O(1)$ as the inverse temperature $β\to \infty$. Meanwhile the rate of leakage away from its centre of mass is $O(e^{-β})$. Futhermore, the quasi-stationary distribution is localised on a length scale of order $O(β^{-\frac12})$. We thus provide a partial answer to a question posed by Carrillo, Craig, and Yao (Aggregation-Diffusion Equations: Dynamics, Asymptotics, and Singular Limits. In: Active Particles, Volume 2, 2019) in the microscopic setting.

Separation of time scales in weakly interacting diffusions

TL;DR

The paper analyzes metastable behavior of weakly interacting Brownian particles on with a localised attractive potential. By projecting out the centre-of-mass and studying the killed, projected diffusion through its generator, the authors connect the droplet state to the quasi-stationary distribution and use spectral analysis to quantify time-scale separation. They establish that the convergence to the droplet (QSD) occurs at rate in inverse temperature , while leakage from the droplet occurs at rate , and that the droplet is localized on length scale . Using a Schrödinger/Witten Laplacian framework and Simon’s norm-resolvent techniques, they obtain low-temperature asymptotics with decaying exponentially and , together with a precise description of multiscale convergence to the QSD and droplet dynamics. The results provide a rigorous microscopic counterpart to the aggregation-diffusion metastability questions and yield a quantitative, multiscale picture of droplet formation and decay.

Abstract

We study metastable behaviour in systems of weakly interacting Brownian particles with localised, attractive potentials which are smooth and globally bounded. In this particular setting, numerical evidence suggests that the particles converge on a short time scale to a "droplet state" which is , i.e. persists on a much longer time scale than the time scale of convergence, before eventually diffusing to . In this article, we provide rigorous evidence and a quantitative characterisation of this separation of time scales. Working at the level of the empirical measure, we show that (after quotienting out the motion of the centre of mass) the rate of convergence to the quasi-stationary distribution, which corresponds with the droplet state, is as the inverse temperature . Meanwhile the rate of leakage away from its centre of mass is . Futhermore, the quasi-stationary distribution is localised on a length scale of order . We thus provide a partial answer to a question posed by Carrillo, Craig, and Yao (Aggregation-Diffusion Equations: Dynamics, Asymptotics, and Singular Limits. In: Active Particles, Volume 2, 2019) in the microscopic setting.

Paper Structure

This paper contains 18 sections, 15 theorems, 133 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation, preliminaries and main results
  3. Assumptions on the potential W
  4. Projected dynamics
  5. The empirical measure
  6. Markov/sub-Markov semigroups and generators
  7. 1. Properties of the Markov semigroup $S_t$.
  8. 2. Properties of the sub-Markov semigroup $P_t$.
  9. 3. Spectrum and heat kernel representation of $P_t$.
  10. One point compactification
  11. The quasi-stationary distribution
  12. Statement of the main results
  13. Proof of exponential asymptotics
  14. Proof of refined asymptotics
  15. Approximating Schrödinger operators on the whole space
  16. ...and 3 more sections

Key Result

Theorem 1.1

For any $N \in \mathbb N$ and large $\beta\gg1$, there exist constants $\lambda_{1,\beta,N}>0$ and $\lambda_{2,\beta,N}>0$ such that $\mu_t^N$ converges to the random droplet configuration $\bar{\mu}^N$ on a time scale of order $\lambda_{2,\beta,N}^{-1}$, while the leakage from the droplet state occ Moreover, the limit $\lambda_{2,N}$ is bounded away from $0$ uniformly in $N$. Finally, $\bar{\mu}^

Figures (2)

  • Figure 1: A schematic depiction of dynamical metastability: the coloured paths represent trajectories of a dynamical system which can be divided into a regime of fast convergence towards the slow submanifold $M$ (red), and slow motion along $M$ (blue).
  • Figure 2: Simulation showing convergence to a single droplet state on a short time scale, followed by the eventual escape of particles from the droplet on a much longer time scale. The simulations are performed with $W=1-e^{-|x|^2}$ and $\beta=0.3$.

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 2.2: Low temperature asymptotics on the exponential scale
  • Remark 2.3
  • Example 2.4
  • Theorem 2.5: Refined low temperature asymptotics
  • Theorem 2.6
  • Corollary 2.7
  • proof
  • Theorem 3.1: M95
  • Lemma 3.2
  • ...and 18 more