Samples the phase space of a constant-P, constant-T ensemble with the
appropriate phase space probability. In addition to the particle
moves performed at constant $V,T$, changes of the volume
$V$ are attempted and accepted/rejected according to a detailed
balance scheme similar to the original NVT-Metropolis procedure.
A simple LJ simulation of the NPT type is sketched here:
For given $T$ and $P$, let the instantaneous volume be
$V$ and the particle positions $\vec{r}_{i}$, $i=1 \dots N$.
Choose a particle $i$
and perform a trial move $\vec{r}_{i} \rightarrow \vec{r'}_{i}$
in the usual manner.
Compute the energy difference between the trial configuration and
the given one:
$\Delta U= U(\vec{r'}_{i})+U(\vec{r}_{i})$, where
$U(\vec{r}_{i}) = \sum_{j \neq i} U_{ij}$
etc.
Accept or reject the trial configuration with probability
$
P_{acc}^{move}=\min\{1, \exp^{- \beta \Delta U }\}
$
Repeat this basic MC step for a number of particles; usually all $N$
particles are treated in sequence.
Now perform a trial volume change $V \rightarrow V' = V+ \Delta V$;
all particle coordinates are (implicitely) scaled by $f = (V'/V)^{1/3}$,
which entails a change in energy of $\Delta U$.
Compute the enthalpy change $\Delta H = \Delta U + P \Delta V$.
The relative probability of the trial state as compared to the
current one is then
$
w=(V'/V)^{N} \, \exp\{-\Delta H \}
$
or
$
w=\exp{- \beta (\Delta H - kT N \ln (V'/V))}
$
In the spirit of Metropolis' asymmetric rule, we
accept/reject the new volume according to
As usual, the maximum particle step and the maximum volume change are
adjusted to achieve an acceptance ratio near
$0.5$.
Note 1: If you are lucky, the model pair potential may be
written in the scalable form
$u(r_{ij}) = f^{m} u(s_{ij})$
where $s_{ij}$ is a scaled distance, and $f$
is the scaling factor. In such cases the total potential energy after
a volume change need not be recalculated from scratch; rather, we have
$U'_{pot}=f^{m} \sum_{i<j}u(s_{ij})=f^{m} U_{pot}$.
As an example, take the $r^{-12}$
term in the Lennard-Jones potential. When scaling all distances
from
$r_{ij}$ to $f r_{ij}$, where
$f \equiv (V'/V)^{1/3}$, we have
Phase separations are best studies with this method. Two separate
boxes are used, and in addition to particle moves and volume changes
a transfer of particles between the boxes is permitted.
To achieve constant and equal temperature, pressure, and chemical
potential in both boxes the following procedure (due to Panagiotopoulos)
is used:
Let $V_{1,2}$ be the box volumes, with a constant
$V=V_{1}+V_{2}$. The particle numbers are
$N_{1,2}$, again with a constant total of
$N=N_{1}+N_{2}$.
Choose a particle
$i$
in one of the boxes and perform a trial move
$\vec{r}_{i} \rightarrow \vec{r'}_{i}$
in the usual manner.
Compute the energy difference between the trial configuration and
the given one:
$\Delta U= U(\vec{r'}_{i})+U(\vec{r}_{i})$, where
$U(\vec{r}_{i}) = \sum_{j \neq i} U_{ij}$
etc.
Accept or reject the trial configuration with probability
$
P_{acc}^{move}=min\{1, \exp^{- \beta \Delta U }\}
$
Repeat this basic MC step for a number of particles; usually all
$N$
particles in both boxes are treated in sequence.
Now perform a trial volume change
$V_{1} \rightarrow V_{1}' = V_{1}+ \Delta V$; since the total volume
is conserved the volume $V_{2}$ must change by $-\Delta V$.
In each box $m=1,2$
all particle coordinates are (implicitely) scaled by
$f = (V'_{m}/V_{m})^{1/3}$,
which entails a change in energy of
$\Delta U_{m}$.
Similar to NPT, the relative probability of the trial state
is related to the quantity
Choose one of the boxes $m=1,2$
with equal probabilites. Choose any of
the particles in box $m$, remove it, and place it at an arbitrary position
in the other box, $m'$. The total Gibbs potential then changes according to
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,
we now subject it to 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
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