Fundamentals of serial generalized-ensemble methods

SGE methods deal with a set of $N$ ensembles associated with different dimensionless Hamiltonians $h_n(x,p)$, where $x$ and $p$ denote the atomic coordinates and momenta of a microstate^6.1, and $n=1, 2, \dots, N$ denotes the ensemble. Each ensemble is characterized by a partition function expressed as

$$Z_n = \int {\rm e}^{-h_n(x,p)} \ {\rm d}x \ {\rm d}p.$$ (6.1)

In ST simulations we have temperature ensembles and therefore the dimensionless Hamiltonian is

$$h_n(x,p) = \beta_n H(x,p),$$ (6.2)

where $H(x,p)$ is the original Hamiltonian and $\beta_n = (k_B T_n)^{-1}$, with $k_B$ being the Boltzmann constant and $T_n$ the temperature of the $n$th ensemble. If we express the Hamiltonian as a function of $\lambda$, namely a parameter correlated with an arbitrary collective coordinate of the system (or even corresponding to the pressure), then the dimensionless Hamiltonian associated with the $n$th $\lambda$-ensemble is

$$h_n(x,p) = \beta H(x,p; \lambda_n).$$ (6.3)

Here all ensembles have the same temperature. It is also possible to construct a generalized ensemble for multiple parameters[120] as

$$h_{n l}(x,p) = \beta_n H(x,p; \lambda_l).$$ (6.4)

In this example two parameters, $T$ and $\lambda$, are employed. However no restraint is actually given to the number of ensemble-spaces. Generalized-ensemble algorithms have a different implementation dependent on whether the temperature is included in the collection of sampling spaces (Eqs. 6.2 and 6.4) or not (Eq. 6.3). Here we adhere to the most general context without specifying any form of $h_n(x,p)$.

In SGE simulations, the probability of a microstate $(x,p)$ in the $n$th ensemble [from now on denoted as $(x,p)_n$] is proportional to $\exp[ -h_n(x,p) + g_n]$, where $g_n$ is a factor, different for each ensemble, that must ensure almost equal visitation of the $N$ ensembles. The extended partition function of this ``system of ensembles'' is

$$Z = \sum_{n=1}^N \int {\rm e}^{-h_n(x,p) + g_n} \ {\rm d}x \ {\rm d}p = \sum_{n=1}^N Z_n {\rm e}^{g_n},$$ (6.5)

where $Z_n$ is the partition function of the system in the $n$th ensemble (Eq. 6.1). In practice, SGE simulations work as follows. A single simulation is performed in a specific ensemble, say $n$, using Monte Carlo or molecular dynamics sampling protocols, and after a certain interval, an attempt is made to change the microstate $(x,p)_n$ to another microstate of a different ensemble, $(x^\prime, p^\prime)_m$. Since high acceptance rates are obtained as the ensembles $n$ and $m$ overlap significantly, the final ensemble $m$ is typically close to the initial one, namely $m = n \pm 1$^6.2. In principle, the initial and final microstates can be defined by different coordinates and/or momenta ( $x \ne x^\prime$ and/or $p \ne p^\prime$), though the condition $x = x^\prime$ is usually adopted. The transition probabilities for moving from $(x,p)_n$ to $(x^\prime, p^\prime)_m$ and vice versa have to satisfy the detailed balance condition

$$P_n(x,p) P(n \rightarrow m) = P_m(x^\prime,p^\prime) P(m \rightarrow n),$$ (6.6)

where $P_n(x,p)$ is the probability of the microstate $(x,p)_n$ in the extended canonical ensemble (Eq. 6.5)

$$P_n(x,p) = Z^{-1} {\rm e}^{-h_n(x,p) +g_n}.$$ (6.7)

In Eq. 6.6, $P(n \rightarrow m)$ is a shorthand for the conditional probability of the transition $(x,p)_n \rightarrow (x^\prime, p^\prime)_m$, given the system is in the microstate $(x,p)_n$ [with analogous meaning of $P(m \rightarrow n)$]. Using Eq. 6.7 together with the analogous expression for $P_m(x^\prime,p^\prime)$ in the detailed balance and applying the Metropolis's criterion, we find that the transition $(x,p)_n \rightarrow (x^\prime, p^\prime)_m$ is accepted with probability

$${\rm acc}[n \rightarrow m] = \min(1, {\rm e}^{h_n(x, p) - h_m(x^\prime, p^\prime) + g_m - g_n}).$$ (6.8)

The probability of sampling a given ensemble is

$$P_n = \int P_n(x,p) \ {\rm d}x \ {\rm d}p = Z_n \ Z^{-1} \ {\rm e}^{g_n}.$$ (6.9)

Uniform sampling sets the condition $P_n= N^{-1}$ for each ensemble ( $n=1, \dots, N$), that leads to the equality

$$g_n = - \ln Z_n + \ln \left( \frac{Z}{N} \right).$$ (6.10)

Equation 6.10 implies that, to get uniform sampling, the difference $g_m - g_n$ in Eq. 6.8 must be replaced with $f_m - f_n$, where $f_n$ is the dimensionless free energy related to the actual free energy of the ensemble $n$ by the relation $f_n = \beta F_n = -\ln Z_n$, where $\beta$ is the inverse temperature of the ensemble. Here we are interested in determining such free energy differences that will be referred as optimal weight factors, or simply, optimal weights. Accordingly, in the acceptance ratio we will use $f_n$ instead of $g_n$.