6.1 NPT Monte Carlo

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 an evaluation of the enthalpy change. 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
  

  $$P_{acc}^{move}=min\{1, \exp^{- \beta \Delta U }\}$$ (6.1)

  
  

• 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
  

  $$\Delta H = \Delta U + P \Delta V - kT N ln (V'/V)$$ (6.2)

  
  
  and accept/reject the new volume according to
  

  $$P_{acc}^{breathe}=min\{1, \exp^{-\beta \Delta H} \}$$ (6.3)

  
  

• This ends the MC cycle.

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

$$U'_{rep}=4 \epsilon \sum_{j>i} f^{-12} r_{ij}^{-12} = ... ...psilon f^{-12} \sum_{j>i} r_{ij}^{-12} = f^{-12} U_{rep}$$ (6.4)

Note 2: A sample program for NPT MC (and for many other simulation techniques), may be found on the web page of Allen and Tildesley's textbook, www.ccl.net/cca/software/SOURCES/FORTRAN/allen-tildesley-book

Note 3: A JAVA applet for NPT MC (and for many other simulation techniques), may be found on the web page of David Kofke, U of Buffalo, www.cheme.buffalo.edu/courses/ce530/Applets/applets.html