Quantitative Metastability in the Linear Becker-Döring Equations

par Inmaculada Benítez Berral

mardi 29 sept. 2026, 11:15 → 12:15 Europe/Paris

The Becker-Döring equations describe the dynamics of cluster formation and have long been known to exhibit metastable behaviour near the critical monomer concentration. In the linear setting, where the monomer concentration is fixed, Penrose [3, 2] and Kreer [1] established fundamental results on the metastable regime and the long-time behaviour of solutions.

In this talk, we revisit this metastable behaviour from a quantitative perspective. We study the convergence of solutions to equilibrium in the subcritical and supercritical regimes and obtain explicit estimates for the exponential relaxation rate. Our approach combines entropy methods with discrete Hardy inequalities and relates the convergence rate to the spectral properties of the associated linear operator.

We show that, as the critical concentration is approached, the relaxation rate becomes small, providing a quantitative description of the metastable time scale. In the supercritical regime, we also identify the role of the critical cluster size and the exponentially small flux through it, which provides a natural interpretation of the slow relaxation.

References

[1] Markus Kreer. Classical Becker-Döring cluster equations: Rigorous results on metastability and long-time behaviour. Annalen der Physik, 505(4):398–417, 1993.

[2] Oliver Penrose. Metastable states for the Becker-Döring cluster equations. Communications in Mathematical Physics, 124(4):515–541, 1989.

[3] Oliver Penrose and Joel L. Lebowitz. Towards a rigorous molecular theory of metastability. Fluctuation Phenomena, 7:293–340, 1987.