Expanded Ensemble¶

In an expanded ensemble simulation 68, both the coordinates and the thermodynamic ensemble are treated as configuration variables that can be sampled over. The probability of any given state can be written as:

(137)¶\[P(\vec{x},k) \propto \exp\left(-\beta_k U_k + g_k\right),\]

where \(\beta_k = \frac{1}{k_B T_k}\) is the \(\beta\) corresponding to the \(k\)th thermodynamic state, and \(g_k\) is a user-specified weight factor corresponding to the \(k\)th state. This space is therefore a mixed, generalized, or expanded ensemble which samples from multiple thermodynamic ensembles simultaneously. \(g_k\) is chosen to give a specific weighting of each subensemble in the expanded ensemble, and can either be fixed, or determined by an iterative procedure. The set of \(g_k\) is frequently chosen to give each thermodynamic ensemble equal probability, in which case \(g_k\) is equal to the free energy in non-dimensional units, but they can be set to arbitrary values as desired. Several different algorithms can be used to equilibrate these weights, described in the mdp option listings.

In GROMACS, this space is sampled by alternating sampling in the \(k\) and \(\vec{x}\) directions. Sampling in the \(\vec{x}\) direction is done by standard molecular dynamics sampling; sampling between the different thermodynamics states is done by Monte Carlo, with several different Monte Carlo moves supported. The \(k\) states can be defined by different temperatures, or choices of the free energy \(\lambda\) variable, or both. Expanded ensemble simulations thus represent a serialization of the replica exchange formalism, allowing a single simulation to explore many thermodynamic states.