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6.3 Extended Gibbs ensemble MC

At higher densities the Gibbs ensemble MC method is plagued by a low acceptance probability of particle insertion. To overcome this problem, several authors suggested to combine the GEMC method with a kind of ``scaled particle'', or ``extended ensemble'' strategy. In the following we will describe the procedure developed by Strnad and Nezbeda (1999).

The basic idea of the extended ensemble is that in addition to the states where a box contains $N$ or $N+1$ particles, there may be a total of $k$ states in which one particle is incompletely coupled to the system, having a smaller size, or potential coupling parameter $\sigma_{i}, i=1...k$. For a complete definition of the extended ensemble, weights have to be assigned, in an arbitrary manner, to the intermediate box states. In the original work of Strnad and Nezbeda there was only one intermediate state ($k=1$), and the corresponding weight was set to $w_{1}=1$.

Generally, if all weights $w_{i}, i=1...k$ are taken to be equal, they cancel from the pertinent formulae. To keep things simple, we will therefore assume equal weights. For the same reason we will take all trial probabilities $p_{i,i \pm 1}=1/2$. Note, however, that the efficiency of the method may be much enhanced by using non-uniform weights $w_{i}$.

Instead of transferring a particle from box $m$ to $m'$ in a single step, it now undergoes a shrinking process through the $k$ intermediate sizes before it is transferred to $m'$.

Strnad and Nezbeda suggested two possible implementations of their method, denoted as EGE1 and EGE2:

EGE1: Particle $i$ in box $m$ is first shrunk to its smallest size $k$, then transferred to box $m'$, to be re-inflated there. Let the energy difference in box $m$ between the states $i+1$ and $i$ of the scaled particle be $\Delta U_{m,i,i+1} \equiv U_{m,i+1}-U_{m,i}$ etc. The acceptance probability of a decoupling/coupling step is then
P_{acc}^{scale}=\min\{ 1, \exp{(-\beta \Delta U_{m,i,i+1})}\}
\end{displaymath} (6.10)

As soon as $i=k$, the transfer from $m$ to $m'$ is accepted with probability
P_{acc}^{trans}=\min\{ 1,
\exp{\left(-\beta \left(U_{m',k}-U_{m,k}\right) \right)}\}
\end{displaymath} (6.11)

where $U_{m',k}$ is the energy in box $m'$ upon insertion of a scaled particle in state $k$.

We expect that the latter acceptance probability, which is so small in the basic Gibbs ensemble MC, will be larger since only a minuscule new particle is inserted in $m'$.

EGE2: A particle in box $m$ is shrunk simultaneously with an inflation of another particle in box $m'$.

In a sample computation, Strnad and Nezbeda report that EGE2 shows no higher efficiency than EGE1.

[to be completed... March 02]

Lit.: Strnad and Nezbeda, Mol.Simul. 22 (1999) 183
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F. J. Vesely / University of Vienna