# Comparison of computational complexities of MD versus MC simulations

In my humble understanding MD simulations of systems with short-range(like LJ interactions) and long-range(electrostatic) has a computational complexity $$O(N . log(N))$$. What will be the computational complexity of a MC(monte-carlo) for the same system? Also what will be the complexities for both cases if electrostatic computations are neglected?

• The answer should be the same, For short range only this is pretty straightforward to see, and is O(N). Electrostatics is harder (and technically can also be done in O(N) but rarely is). I'll leave a proper answer to somebody who does MC as after 5 seconds thought I can't see a quick way to justify it being the same as MD for long range terms - I really only do MD. – Ian Bush May 6 at 6:04
• @IanBush : So the real advantage of MC is not in its fast algorithm, but the sampling of wider configurational space? – dexterdev May 6 at 10:50
• If ergodicity holds they both sample the same space - but may do with differing degrees of effectiveness – Ian Bush May 6 at 12:29
• Oh, and just to be absolutely clear note a single move in MC typically (but not always) move just one particle while MD they all move. So you have to compare N steps of MC with 1 step of MD. The complexity should be the same. – Ian Bush May 6 at 12:30