  Parallel tempering, also known as replica exchange Monte Carlo, is studied in
the context of two simple free energy landscapes. The first is a double well
potential defined by two macrostates separated by a barrier. The second is a
`golf course' potential defined by microstates having two possible energies
with exponentially more high energy states than low energy states. The
equilibration time for replica exchange is analyzed for both systems. For the
double well system, parallel tempering with a number of replicas that scales as
the square root of the barrier height yields exponential speedup of the
equilibration time. On the other hand, replica exchange yields only marginal
speed-up for the golf course system. For the double well system, the free
energy difference between the two wells has a large effect on the equilibration
time. Nearly degenerate wells equilibrate much more slowly than strongly
asymmetric wells. It is proposed that this difference in equilibration time may
lead to a bias in measuring overlaps in spin glasses. These examples illustrate
the strengths and weaknesses of replica exchange and may serve as a guide for
understanding and improving the method in various applications.
