Scaling of plain hybrid Monte Carlo

The object of using molecular dynamics as the trial step in Monte Carlo is to allow large steps in exploring configuration space while maintaining a reasonable acceptance ratio. But the acceptance ratio will decrease dramatically when the system size increases. To maintain a certain acceptance ratio, the step size must decrease as the system size increases.

Note that the acceptance ratio, $\min(1,e^{-\delta H})$, will be unity if the molecular dynamics conserve Hamiltonian exactly, that is, $\delta H=0$. It is impossible to integrate Hamilton's equation exactly for the forces used in MD. But the acceptance ratio will be improved if the discrete integrator conserves the Hamiltonian with higher accuracy. The cost of using higher accuracy integrators for MD offsets potential gains, except at very high problem sizes.