# Finite-size effects in molecular dynamics simulation

I will be blunt: this is an old exam question of a course that covers Molecular Dynamics (MD) and Monte Carlo (MC) simulations.

Explain how to determine the severity of finite-size effects.

My idea: Since we are talking about MD and MC, finite-size effects are related to the fact that we can only simulate systems of finite sizes. Contrarily, in nature systems usually approach very large sizes, i.e. they are in the thermodynamic limit.

I found these lecture slides which gives me some ideas, but I am not sure that this is what is being asked in the question above.

Any thoughts, and pointers to slides / scripts are very welcome.

Periodic boundary conditions (PBC) are a common technique to get closer to the thermodynamic limit while keeping the computation feasible. You could set up an experiment simulating a box of argon with and without PBC and study how the radial distribution function g(r) changes in each case, you should observe a difference in g(r) for large values of r.

Another approach could consist of revisiting the derivation of the canonical ensemble (or other ensembles). One way this derivation is typically done is by truncating a Taylor expansion where the remainder is a function of the number of particles N that goes to zero as N goes to infinity. You could consider N fixed and quantify the error of the approximation that way (see, for example, the derivation of equation 2.23 in this paper).

• Thanks for the answer. I think it does not explain what I am looking for though. What I am looking for is basically what happens to the observed quantities in my simulation when I increase the size of the simulation box. I'm not sure if argon is a good example for this. The 2D Ising model is easier to handle, since the number of spins is equal to the number of grid elements N. Which is also the route they are going in the link that I posted. I'm just still hung up on the word severity. I.e., when is something severe or not severe in this context? – seb Aug 23 '13 at 13:48
• Hi Sebastian. The link to the lecture slides was not properly included in the question when I wrote my reply, hence my suggestion of a box of Argon (as it is a simple but not completely trivial model). Based on a cursory look of the slides, I'd take a careful look at the section "How to get reliable error estimates in Finite Size Scaling?" and figure out when the errors are expected to be large due to small system size. Other than that, both of my suggestions above are relevant to you question in general. – Juan M. Bello-Rivas Aug 23 '13 at 19:48
• thanks Juan, my bad. I think something in the notation how to include links has changed at SE – seb Aug 25 '13 at 8:40
• I marked your answer as accepted answer. Since your comment gave me the right clue. Thanks. – seb Aug 25 '13 at 12:32
• You're welcome. I'm glad to be of help. – Juan M. Bello-Rivas Aug 25 '13 at 13:21

Presupposing an observable of interest, simulate over different finite size scales (with whatever initial and boundary conditions interest you), and demonstrate that the size scale affects the values of the estimate of the observable (taking care to converge the estimate with respect to sampling). In the limit of a sufficiently large size scale, you can see the estimate of the observable approach the true value for that model, and thus measure the severity of finite-size effects.

You can simulate a range of systems varying in size, from smaller to larger, and show that a given property of interest mades a plato as system size increases (note that You'll have to equilibrate longer, as the size of the system grows).