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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\beta \rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. Meanwhile the rate of leakage away from its centre of mass is <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$O(e^{-\beta })$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>β</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Furthermore, the quasi-stationary distribution is localised on a length scale of order <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$O(\beta ^{-\frac{1}{2}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>β</mi> <mrow> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Our proofs rely on understanding the large <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation>-asymptotics of the first two eigenvalues of the generator, which we study using techniques from semiclassical analysis. We thus provide a partial answer to a question posed by Carrillo et al. (see Aggregation–diffusion equations: dynamics, asymptotics, and singular limits. Active particles. Advances in theory, models, and applications, modeling and simulation in science, engineering and technology, vol 2, pp 65–108, Birkhäuser/Springer, Cham, 2019, Section 3.2.2) in the microscopic setting.

Separation of Time Scales in Weakly Interacting Diffusions

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2026年4月30日

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

摘要

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 $\beta \rightarrow \infty $ $β \to \infty$ . Meanwhile the rate of leakage away from its centre of mass is $O(e^{-\beta })$ $O (e^{- β})$ . Furthermore, the quasi-stationary distribution is localised on a length scale of order $O(\beta ^{-\frac{1}{2}})$ $O (β^{- \frac{1}{2}})$ . Our proofs rely on understanding the large $\beta $ $β$ -asymptotics of the first two eigenvalues of the generator, which we study using techniques from semiclassical analysis. We thus provide a partial answer to a question posed by Carrillo et al. (see Aggregation–diffusion equations: dynamics, asymptotics, and singular limits. Active particles. Advances in theory, models, and applications, modeling and simulation in science, engineering and technology, vol 2, pp 65–108, Birkhäuser/Springer, Cham, 2019, Section 3.2.2) in the microscopic setting.