# Separation of degrees of freedom in Monte Carlo simulation

My question is probably very simple and deals with separation of degrees of freedom in a Monte Carlo Brownian dynamics simulation.

Dealing with a particle in an external potential, I want to simulate a Brownian motion in three spatial dimensions. Is it possible to evaluate the individual degrees of freedom separately i.e. calculate displacement in x, evaluate the probabilities, and accept/reject the step and then repeat for y and z? Or do I have to evaluate all degrees of freedom in a single MC step?

Now what if the degrees of freedom are not independent, i.e. the displacement in x will change the potential (probability) in y etc.? Can the individual degrees of freedom be still calculated separately? Let's say I calculate displacement in x which gets accepted but displacement in y gets rejected. Do I have to throw out both of the displacements? Or can I keep the displacement in x?

I understand that in multi-particle systems, the MC method doesn't evaluate all particles at the same time, therefore it should be possible to evaluate the degrees of freedom separately even if they influence the probability distribution of other degrees of freedom.

• This may get a better answer at Computational Science
– Colin McFaul
Sep 30 '13 at 23:33

## 3 Answers

I assume you are writing a Metropolis-Hastings Monte Carlo algorithm. So, you are trying to generate moves to sample the configuration space and accept/reject these moves based on energy considerations (MHMC), right? If so, then you are free to define your moves however you choose. There comes eventually the question of efficiency, as proposing unlikely moves will result in small acceptance probabilities over simulation time.

As per your question about a displacement in x- and y-, I suppose this depends upon your method of proposing/testing moves. Generally, we do not sample configuration space this way, you should just propose one move (say -5 Angstroms in the y direction and 2 Angstroms in the x direction), test whether it should be accepted or rejected based on the MH algorithm, and move on without breaking it down into its components. This is really a matter of efficiency and you are free to sample as you see fit as long as you preserve ergodicity and detailed balance.

Note that it is also perfectly acceptable to grab a few atoms (or molecules, clusters, whatever you are studying) and move them all at once and then compute the acceptance probability.

Leaving efficiency considerations aside, your moves are OK if they satisfy detailed balance.

What you describe is commonly called "Gibbs sampling", see http://en.wikipedia.org/wiki/Gibbs_sampler