Separated relaxation scales from a condensate-profile bottleneck

A translation can change the coherence between two Fourier modes of a condensate while leaving the distribution of its field values intact. We study the relaxation of these signals in a spherical lattice gas whose local stochastic updates conserve the field sum and squared norm. At fixed interaction strength, the mixed Fourier coherence has correlation times of order T−2 and the modal concentration has times of order T−3, with a common factor Nn2 for N sites and lattice side n. Both observables have nonvanishing equilibrium variance, and the bounds apply to integrated and fixed-fraction correlation times. In the variational proof, corrections for the two conserved densities cancel the contribution linear in thermal fluctuations for observables of the value distribution. Quadratic fluctuations give the slower scale. A bounded tangent current gives the matching upper bound on the faster correlation time. The bounds hold at sufficiently large volume for each fixed low temperature T. This preprint includes the full derivations in Appendixes A–I. The source archive contains the LaTeX manuscript and the deterministic quadrature record and plotting code for Figure 1.