  We show that the acceptance probability for swaps in the parallel tempering
Monte Carlo method for classical canonical systems is given by a universal
function that depends on the average statistical fluctuations of the potential
and on the ratio of the temperatures. The law, called the incomplete beta
function law, is valid in the limit that the two temperatures involved in swaps
are close to one another. An empirical version of the law, which involves the
heat capacity of the system, is developed and tested on a Lennard-Jones
cluster. We argue that the best initial guess for the distribution of
intermediate temperatures for parallel tempering is a geometric progression and
we also propose a technique for the computation of optimal temperature
schedules. Finally, we demonstrate that the swap efficiency of the parallel
tempering method for condensed-phase systems decreases naturally to zero at
least as fast as the inverse square root of the dimensionality of the physical
system.
