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
   

   $$\delta x = \Delta R \xi_x$$ (84)

   
   
   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.

Details of the rigid body rotation implementation are presented in the text, in Sec. 3.3.2. The two techniques described are vector addition for linear molecules, and quaternions for nonlinear molecules. For the purposes of one of our simple 2D case-studies, we will consider rotation by a small angle, which is effectively equivalent to the vector addition method described by Frenkel and Smit.