Relaxation of negative-temperature states in the discrete nonlinear Schrodinger chain · revision history

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Title

Relaxation of negative-temperature states in the discrete nonlinear Schrodinger chain

Plain

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

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 $N$.

Settled by

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.

Kind

well-posed

Answer type

value

References

(empty)

Literature status

partially-resolved

Status note

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.

Posed since

2000