  Ion traps are a versatile tool to study nonequilibrium statistical physics,
due to the tunability of dissipation and nonlinearity. We propose an experiment
with a chain of trapped ions, where dissipation is provided by laser heating
and cooling, while nonlinearity is provided by trap anharmonicity and beam
shaping. The collective dynamics are governed by an equation similar to the
complex Ginzburg-Landau equation, except that the reactive nature of the
coupling leads to qualitatively different behavior. The system has the unusual
feature of being both oscillatory and excitable at the same time. We account
for noise from spontaneous emission and find that the patterns are observable
for realistic experimental parameters. Our scheme also allows controllable
experiments with noise and quenched disorder.
