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Long-time propagation of chaos and exit times for metastable mean-field particle systems

Pierre Monmarché

TL;DR

This work addresses the long-time behavior of mean-field particle systems in metastable regimes by introducing a modified dynamics with a uniform-in-$N$ log-Sobolev inequality (LSI). The authors prove that exit times from metastable basins grow exponentially with the number of particles and that the empirical measure remains close to an autonomous nonlinear McKean–Vlasov process up to exit. They establish a practical criterion for uniform LSI (Theorem LSIN) and derive long-time convergence and exponential exit-time results (Theorems 1 and 2) for quadratic interactions, supported by a detailed construction of a modified energy yielding global LSI. Uniform LSIs enable propagation-of-chaos and nonlinear functional-inequality consequences for the mean-field limit, and extend to kinetic dynamics and discretizations. The work lays groundwork for rigorous Arhenius-type laws in infinite-dimensional mean-field settings and suggests directions for extending the theory to more general interactions and processes.

Abstract

Systems of stochastic particles evolving in a multi-well energy landscape and attracted to their barycenter is the prototypical example of mean-field process undergoing phase transitions: at low temperature, the corresponding mean-field deterministic limit has several stationary solutions, and the empirical measure of the particle system is then expected to be a metastable process in the space of probability measures, exhibiting rare transitions between the vicinity of these stationary solutions. We show two results in this direction: first, the exit time from such metastable domains occurs at time exponentially large with the number of particles and follows approximately an exponential distribution; second, up to the expected exit time, the joint law of particles remain close to the law of independent non-linear McKean-Vlasov processes.

Long-time propagation of chaos and exit times for metastable mean-field particle systems

TL;DR

This work addresses the long-time behavior of mean-field particle systems in metastable regimes by introducing a modified dynamics with a uniform-in- log-Sobolev inequality (LSI). The authors prove that exit times from metastable basins grow exponentially with the number of particles and that the empirical measure remains close to an autonomous nonlinear McKean–Vlasov process up to exit. They establish a practical criterion for uniform LSI (Theorem LSIN) and derive long-time convergence and exponential exit-time results (Theorems 1 and 2) for quadratic interactions, supported by a detailed construction of a modified energy yielding global LSI. Uniform LSIs enable propagation-of-chaos and nonlinear functional-inequality consequences for the mean-field limit, and extend to kinetic dynamics and discretizations. The work lays groundwork for rigorous Arhenius-type laws in infinite-dimensional mean-field settings and suggests directions for extending the theory to more general interactions and processes.

Abstract

Systems of stochastic particles evolving in a multi-well energy landscape and attracted to their barycenter is the prototypical example of mean-field process undergoing phase transitions: at low temperature, the corresponding mean-field deterministic limit has several stationary solutions, and the empirical measure of the particle system is then expected to be a metastable process in the space of probability measures, exhibiting rare transitions between the vicinity of these stationary solutions. We show two results in this direction: first, the exit time from such metastable domains occurs at time exponentially large with the number of particles and follows approximately an exponential distribution; second, up to the expected exit time, the joint law of particles remain close to the law of independent non-linear McKean-Vlasov processes.

Paper Structure

This paper contains 23 sections, 32 theorems, 197 equations, 6 figures.

Table of Contents

  1. Overview
  2. Classical convergence results.
  3. Metastability.
  4. Our approach : a modified process.
  5. Main contributions and organization.
  6. Main results with a quadratic interaction
  7. Settings
  8. Results
  9. Proofs of the main results
  10. Uniform LSI
  11. Consequences of the uniform LSI
  12. Overdamped Langevin dynamics
  13. Other processes
  14. Exponentiality of exit times
  15. The modified energy
  16. ...and 8 more sections

Key Result

Proposition 1

Under Assumption assu:main-result, there exist $a,\eta>0$ such that, for all $N \geqslant 1$, $t_N \geqslant \eta e^{a N}$.

Figures (6)

  • Figure 1: Mean of the stationary solutions as a function of $\sigma$.
  • Figure 2: Fast convergence to stationary solutions, for $\sigma=0.5$ and $N=10^4$ particles initialized independently with $\mathcal{N}(m_0,1/4)$ with $m_0=1$ (in blue) or $m_0=-1$ (in orange). (Left) At time $T=10$, histogram of the particles superimposed with the graph of $\mu_-$ and $\mu_+$. (Right). Trajectory $t\mapsto \bar{X}_t$ up to time $T=1000$.
  • Figure 3: Same as Figure \ref{['fig:fast']} except $N=10^5$.
  • Figure 4: Same as Figure \ref{['fig:fast']} except $N=10^3$, $T=10^3$ and $\sigma=0.8>\sigma_c$.
  • Figure 5: A metastable transition. Same initial conditions as in Figure \ref{['fig:fast']}, but $\sigma=0.64$ and $N=10^3$. (Top) Trajectory $t\mapsto \bar{X}_t$ up to time $T=10^4$. (Bottom) Histogram of the particles superimposed with the graph of $\mu_-$ and $\mu_+$, (Left) at time $T=5000$ before the transition and (Right) at time $T=10^4$ after the transition.
  • ...and 1 more figures

Theorems & Definitions (63)

  • Example 1
  • Example 2
  • Proposition 1
  • Theorem 2
  • Theorem 3
  • proof : Proof of Proposition \ref{['prop:tN']}
  • proof : Proof of Theorem \ref{['thm:main1']}
  • proof : Proof of Theorem \ref{['thm:mainExit']}
  • Theorem 4
  • Lemma 5
  • ...and 53 more