SGE simulations in temperature-space (simulated tempering) and its implementation in the ORAC program

In SGE Monte Carlo simulations conducted in temperature-space (ST simulations), Eq. 6.2 holds. Specifically, since only configurational sampling is performed, we have

$$h_n(x) = \beta_n V(x),$$ (6.11)

where $V(x)$ is the energy of the configuration $x$. Exploiting Eq. 6.11 into Eq. 6.8, we find that transitions from $n$ to $m$-ensemble, realized at fixed configuration, are accepted with probability

$${\rm acc}[n \rightarrow m] = \min(1, {\rm e}^{( \beta_n - \beta_m) V(x) + f_m - f_n}).$$ (6.12)

When the system evolution is performed with molecular dynamics simulations, the situation is slightly more complicate. Suppose to deal with canonical ensembles (to simplify the treatment and the notation we consider constant-volume constant-temperature ensembles, though extension to constant-pressure constant-temperature ensembles is straightforward). Usually, constant temperature is implemented through the Nosé-Hoover method[121,122] or extensions of it[123]. With the symbol $p_t$, we will denote the momentum conjugated to the dynamical variable associated with the thermostat. Also in this case Eq. 6.2 holds, but it takes the form

$$h_n(x,p,p_t) = \beta_n H(x,p,p_t).$$ (6.13)

In this equation, $H(x,p,p_t) = V(x) + K(p) + K(p_t)$ is the extended Hamiltonian of the system, where $V(x)$ is the potential energy, while $K(p)$ and $K(p_t)$ are the kinetic energies of the particles and thermostat, respectively. As in Monte Carlo version, transitions from $n$ to $m$-ensemble are realized at fixed configuration, while particle momenta are rescaled as

$$\begin{tabular}{l} $p^\prime = p \ (T_m/T_n)^{1/2}$\ \\ $p^\prime_t = p_t \ (T_m/T_n)^{1/2}$. \end{tabular}$$ (6.14)

As in temperature-REM[124], the scaling drops the momenta out of the detailed balance and the acceptance ratio takes the form of Eq. 6.12. Note that, if more thermostats are adopted[123], then all additional momenta must be rescaled according to Eq. 6.14.

ST is implemented in the ORAC program exactly as it has been done for REM (see Section 5.2). In particular global and local scalings of the potential energy can be realized by keeping fixed the temperature of the system. A generic ensemble $n$ is therefore defined by a coefficient ${\mathbf c}_n$ (see Eq. 5.13) that scales the potential energy ${\mathbf v}(x)$ of the replica (the vectorial form of the potential energy $V(x)$ is used because of possible local scaling), i.e., $V(x) = {\mathbf c}_n \cdot {\mathbf v}(x)$. In this sort of Hamiltonian tempering, the transition from $n$ to $m$-ensemble is accepted with probability

$${\rm acc}[n \rightarrow m] = \min(1, {\rm e}^{ \beta ( {\mathbf c}_n - {\mathbf c}_m ) \cdot {\mathbf v}(x) + f_m - f_n}).$$ (6.15)

In this approach, since the temperature is the same for all ensembles, momentum rescaling (Eq. 6.14) must not be applied. We will see in Section 6.3 how $f_m$ and $f_n$ appearing into Eq. 6.15 are determined.