A ring of BEC pools as a trap for persistent flow
  Mott insulator - superfluid transition in a periodic lattice of Josephson junctions can be driven by tunneling rate increase. Resulting winding numbers $W$ of the condensate wavefunction decrease with increasing quench time in accord with the Kibble-Zurek mechanism (KZM). However, in very slow quenches Bose-Hubbard dynamics rearranges wavefunction phase so that its random walk cools, $\bar{W^2}$ decreases and eventually the wavefunction becomes too cold to overcome potential barriers separating different $W$. Thus, in contrast with KZM, in very slow quenches $\bar{W^2}$ is set by random walk with "critical" step size, independently of $\tau_Q$. 
Jacek Dziarmaga, Marek Tylutki, and Wojciech H. Zurek
cond-mat.quant-gas cond-mat.mes-hall cond-mat.stat-mech quant-ph