  Recent experiments by Sato et al. [1] have explored the dynamics of $^4$He
superflow through an array of nanopores. These experiments have found that, as
the temperature is lowered, phase-slippage in the pores changes its character,
from synchronous to asynchronous. Inspired by these experiments, we construct a
model to address the characteristics of phase-slippage in superflow through
nanopore arrays. We focus on the low-temperature regime, in which the
current-phase relation for a single pore is linear, and thermal fluctuations
may be neglected. Our model incorporates two basic ingredients: (1) each pore
has its own random value of critical velocity (due, e.g., to atomic-scale
imperfections), and (2) an effective inter-pore coupling, mediated through the
bulk superfluid. The inter-pore coupling tends to cause neighbours of a pore
that has already phase-slipped also to phase-slip; this process may cascade,
creating an avalanche of synchronously slipping phases. As the temperature is
lowered, the distribution of critical velocities is expected to effectively
broaden, owing to the reduction in the superfluid healing length, leading to a
loss of synchronicity in phase-slippage. Furthermore, we find that competition
between the strength of the disorder in the critical velocities and the
strength of the inter-pore interaction leads to a phase transition between
non-avalanching and avalanching regimes of phase-slippage.
  [1] Sato, Y., Hoskinson, E. Packard, R. E. cond-mat/0605660.
