  For a harmonic oscillator, Nos\'e's single-thermostat approach to simulating
Gibbs' canonical ensemble with dynamics samples only a small fraction of the
phase space. Nos\'e's approach has been improved in a series of three steps: [
1 ] several two-thermostat sets of motion equations have been found which cover
the complete phase space in an ergodic fashion, [ 2 ] sets of single-thermostat
motion equations, exerting "weak control" over both forces and momenta, have
been shown to be ergodic, and [ 3 ] sets of single-thermostat motion equations
exerting weak control over two velocity moments provide ergodic phase-space
sampling for the oscillator and for the rigid pendulum, but not for the quartic
oscillator or for the Mexican Hat potential. The missing fourth step, motion
equations providing ergodic sampling for anharmonic potentials requires a
further advance. The 2016 Ian Snook Prize will be awarded to the author(s) of
the most interesting original submission addressing the problem of finding
ergodic algorithms for Gibbs' canonical ensemble using a single thermostat.
