chapter 4 section 4.5

SCALING

  Application of a standard Monte Carlo algorithm yielded very poor statistics. This outcome is probably caused by the inability of the volume to adjust quickly to particle diameter changes, and vice versa. A remedy might be to introduce a combined volume- and diameter-change Monte Carlo step: while reducing the volume by a factor , we reduce all diameters by a factor . In addition to more directly coupling the diameter and volume changes, this move is appealing because it requires no overlap test. The acceptance probability for this combined move is

where L is the length of the cubic box with volume and are the scaled diameters. Although this method will produce better results, we can improve the scheme even more by scaling both the diameters and the coordinates, thereby permitting (near-) analytic evaluation of the volume integral. Consider the configurational part of the isobaric semigrand partition function

If we now introduce scaled coordinates and diameters and as above the partition function can be written as

The outermost two integrals do not contain any volume dependence. Moreover, the 'weighting function' defined here can be evaluated entirely at every Monte Carlo move. In this way, there is no need for volume sampling at all. Instead, at every move we obtain the average volume by evaluating

In this scaling method there are only two kinds of Monte Carlo moves: the regular particle displacements and the diameter changes. Both changes are made in the scaled variables. The acceptance probabilities are

where is the change in potential energy associated with the move. If a particle's diameter is changed the integral must be reevaluated. Because the exponential in the integral is a cubic polynomial this cannot be done analytically. However, if we write the integral as

we can approximate it accurately by applying the method of steepest descent. Replacing the function by a second-order Taylor expansion around the maximum of yields

This approximation is possible because the function is large (being proportional to N) and it drops quickly away from the maximum.

Figure:

Solid-fluid coexistence pressure as a function of variance of the imposed activity distribution. In the inset the pressure is reduced to to show the limiting behavior.