Generalized ensemble and tempering simulations: A unified view

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.

Figure 1

Sketch of state space for generalized ensemble sampling in the case of broken ergodicity; $X$ denotes any degree of freedom orthogonal to the control parameter (energy).

Figure 2

Sketch of bifurcations and multifurcations in the case of broken ergodicity for parallel tempering; for certain nodes, the system partitions into several disjoint free energy wells.