Changing randomly a unit vector

Question

For studying a spin model on a lattice, I have to generate a random unit vector starting from a pre-exstisting one. There are multiple ways to do it, but the book I use suggests generating a random displacement vector $\Delta\boldsymbol{S}=\Delta S_{max}(p_1,p_2,p_3)$ with $p_i$ three random numbers in $[-1,1]$ and $\Delta S_{max}$ a parameter, then adding the displacement to the original vector and normalizing the sum. What I don't understand is the following. The authors say to check if $|\Delta\boldsymbol{S}|>\Delta S_{max}$ and generate another displacement if this condition is verified. They say that:

This latter step is necessary to insure that the change in a spin direction is symmetrically distributed around the current spin direction.

Here, the current spin is the pre-existing unit vector. I don't see what they mean with the sentence quoted basically. Why do I need to do that check?

The book is "Introduction to Computer Simulation Methods" by Harvey Gould, Jan Tobochnik, and Wolfgang Christian.

Answer 1

What you’re seeing is a basic acceptance-rejection method. $\Delta S$ generated in that way will be uniform in the ball of radius $\Delta S_{\text{max}}$ centered at the origin.

Added

I imagine you're doing some kind of optimization by random search, in which case the goal is to find a set of spins that extremize some objective function. The way it looks is that you have an initial spin, $S$, and want to explore the improvement a random variation of it yields. In this case $\Delta S_{\text{max}}$ give you a sort of maximum step size, and the vector $S + \Delta S$, with $\Delta S$ generated as you describe, gives you a new candidate after you normalize it. If $\Delta S_{\text{max}}$ is small, say much smaller than 1, the new candidate will be pretty close to the initial one.

So why is ''rotationally symmetric'' important? It seems like a nice thing to have, taking a step isotropically, but I doubt it's important if, in fact, this is an optimization problem. All you really need is a controlled way of generating new, reasonable candidates.