Trial Moves

Particle Displacement. The most common trial move in continuous-space MC is a particle displacement. First, a small number $\Delta R$, representing a maximum displacement, is set. A trial move consists of

1. Randomly select a particle, $i$.
2. Displace x-position coordinate of particle $i$ by a random amount, $\delta x$, which is given by

   $$\delta x = \Delta R \xi_x$$ (90)

   
   
   where $\xi_x$ is a uniform random variate on the interval [-0.5:0.5].
3. Repeat for the $y$ and $z$ coordinates, if applicable.

This move guarantees detailed balance, provided that the random particle selection is uniform; for any given move, selection of all possible particles is equally likely. This means that probability of suggesting a move that displaces a particle, going from a state $n$ to a new state $m$, has the same probability of selecting the same particle while in state $m$ and giving it a displacement that will return the configuration to state $n$. (Do you think such sequential moves ever actually happen?)

For a system of simple particles, random displacements are the only necessary trial moves; thus, $\alpha_{nm}$ is always unity. For more complicated systems, there are zoos of trial moves all over the literature. We will consider some more complicated systems and trial moves later in the course; one that we consider next is rigid rotation.

The question at this point is, how does one choose an appropriate value for $\Delta R$? If $\Delta R$ is too small, the system will not explore phase space given a reasonable amount of computational effort. If it is too large, displacements will rarely result in new configurations which will be accepted in a Metropolis MC scheme. So it takes a bit of trial and error to find a good value for $\Delta R$, and the rule of thumb is to set $\Delta R$ such that the average probability of accepting a new configuration during a run is about 30%. This is termed “tuning $\Delta R$ to achieve a 30% acceptance ratio.” We will go through the exercise of determining such an appropriate value for $\Delta R$ for a simple continuous-space system; namely, 2D hard disks confined to a circle.

Rigid rotation. A second common type of trial move is used in systems of more structured molecules than just simple, single-center spheres. Consider a diatomic with a rigid bond length $r_0$. Clearly, attempting to move one of the two members of the diatomic by a random displacement is likely to result in a new bond length with may be significantly different from $r_0$. So, for a system of diatomics, a reasonable set of trial moves would include

1. Small displacement of molecule center of mass; and
2. Small rotation around molecule center of mass.

With more than one kind of move, an attempt to generate a new state must be preceded by a random selection of the trial move. We can weight each kind of move and then use a random number to decide which move to attempt. For example, let's say that we choose that 80% of all trial moves be displacements, and the balance rotations (we will see later whether or not this is a good choice). Prior to an attempted move, we select a uniform random variate, $\xi$, on the interval $[0,1]$. If $\xi < 0.8$, which it will be 80% of the time, we execute a displacement of a randomly chosen molecule; otherwise, we execute a rotation of a randomly chosen molecule.