Relaxation of negative-temperature states in the discrete nonlinear Schrodinger chain
In plain words
A chain of coupled nonlinear wave sites can be given so much energy per particle that its temperature formally becomes negative, and energy then gathers into a few tall localized peaks called breathers. Whether the chain ever settles into equilibrium from there, and how long it takes, is not known.
Precise statement
For the DNLS chain $H = \sum_n [ (\psi_n* \psi_{(n+1)} + \mathrm{c.c.}) + (g/2)\left|\psi_n\right|^4 ]$ with conserved norm density $a$ and energy density $h$ above the infinite-temperature line h_inf = g a^2, determine whether isolated dynamics from generic initial data relaxes to the microcanonical equilibrium (one breather holding a macroscopic fraction of the norm on an infinite-temperature background), and give the scaling of the relaxation time with chain length $N$ and with h - h_inf. An answer is the asymptotic law, or a proof that the relaxation time diverges with $N$.
What would settle it
Simulations at increasing N that follow breather and background statistics to the predicted microcanonical state, with a theory of breather growth rates that reproduces the measured times.
Status in the literature
The microcanonical equilibrium above the line was derived in 2021 (Gradenigo, Iubini, Livi, Majumdar, J. Stat. Mech. 023201, arXiv:1910.07461), and an adiabatic invariant of tall breathers was shown to make relaxation extremely slow (Iubini et al., Phys. Rev. Lett. 122, 084102, 2019); the large-N law is not established.