  From the underlying Master equations we derive one-dimensional stochastic
processes that describe generalized ensemble simulations as well as tempering
(simulated and parallel) simulations. The representations obtained are either
in the form of a one-dimensional Fokker-Planck equation or a hopping process on
a one-dimensional chain. In particular, we discuss the conditions under which
these representations are valid approximate Markovian descriptions of the
random walk in order parameter or control parameter space. They allow a unified
discussion of the stationary distribution on, as well as of the stationary flow
across each space. We demonstrate that optimizing the flow is equivalent to
minimizing the first passage time for crossing the space, and discuss the
consequences of our results for optimizing simulations. Finally, we point out
the limitations of these representations under conditions of broken ergodicity.
