An entropic barrier to equilibration in the discrete nonlinear Schrödinger equation: Rigorous bounds for the microcanonical Brownian motion
Abstract
The relaxation of the discrete nonlinear Schrödinger equation is known to slow down dramatically when the energy density ε exceeds a critical value εc, where the system enters a condensed phase. Replacing the Hamiltonian dynamics by Brownian motion on the microcanonical surface of N sites at fixed mass and energy, we prove rigorous upper and lower bounds on the equilibration time τeq, which is governed by the first non-zero eigenvalue λ1 of the Laplacian on this surface. In the condensed phase the equilibration time is stretched exponential, not exponential, in the system size. The upper bounds on λ1 come from trial functions: of one occupation for ε<εc, where the problem reduces to a Sturm-Liouville equation, and of two occupations for ε>εc, where they cross the bottleneck through which the condensate moves between sites. The lower bounds on λ1 come from a Bochner-Reilly argument, with a vector field that compensates the concavity of the boundary. As a result, τeq=O(1) for ε<εc, while for ε>εc, asymptotically in N, (√2−1)√(ε−εc)≤lnτeq/√N≤√(ε−εc). We conjecture that the left-hand bound is sharp and the right-hand one is not.
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BibTeX
@misc{arezzo2026entropic,
title = {{An entropic barrier to equilibration in the discrete nonlinear Schrödinger equation: Rigorous bounds for the microcanonical Brownian motion}},
author = {Claudio Arezzo and Antonello Scardicchio},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02025},
eprint = {2611.02025},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02025},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An entropic barrier to equilibration in the discrete nonlinear Schrödinger equation: Rigorous bounds for the microcanonical Brownian motion
AU - Claudio Arezzo
AU - Antonello Scardicchio
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.02025
UR - https://arxiv.org/abs/2611.02025
AB - The relaxation of the discrete nonlinear Schrödinger equation is known to slow down dramatically when the energy density ε exceeds a critical value ε_c, where the system enters a condensed phase. Replacing the Hamiltonian dynamics by Brownian motion on the microcanonical surface of N sites at fixed mass and energy, we prove rigorous upper and lower bounds on the equilibration time τ_eq, which is governed by the first non-zero eigenvalue λ_1 of the Laplacian on this surface. In the condensed phase the equilibration time is stretched exponential, not exponential, in the system size. The upper bounds on λ_1 come from trial functions: of one occupation for ε<ε_c, where the problem reduces to a Sturm-Liouville equation, and of two occupations for ε>ε_c, where they cross the bottleneck through which the condensate moves between sites. The lower bounds on λ_1 come from a Bochner-Reilly argument, with a vector field that compensates the concavity of the boundary. As a result, τ_eq=O(1) for ε<ε_c, while for ε>ε_c, asymptotically in N, (√2−1)√(ε−ε_c)≤lnτ_eq/√N≤√(ε−ε_c). We conjecture that the left-hand bound is sharp and the right-hand one is not.
ER -