15.4 Algorithms for lattice field theories 489
From this, it follows immediately that the modified detailed balance condition
holds:
ρ(′,P′)
ρ(,P)=
ω,P;′,P′A,P;′,P′
ω′,−P′;,−PA′,P′;,P=exp[Hclass(,P)−Hclass(′,P′)].
(15.72)
We see that without momentum refreshings, the canonical distribution is a sta-
tionary distribution of the Markov process. However, for small time steps in the
leap-frog algorithm, the changes in the classical Hamiltonian are very small, and
convergence will be extremely slow. That is the reason why these steps are com-
bined with momentum refreshings, which are compatible with a canonical invariant
distribution too, but which cause more drastic changes in the energy. This method
is usually called thehybrid Monte Carlo method[18].
The important advantage of this Metropolis-improved MD method is that the time
step of the leap-frog algorithm can be stretched considerably before the acceptance
rate of the Metropolis step drops too low. This causes the correlation time for
the ‘microcanonical’ part, measured in time steps, to be reduced considerably. We
have put the quotes around ‘microcanonical’ because the energy is not conserved
very well with a large time step. If the time step is taken too large, the Verlet method
becomes unstable (seeAppendix A7.1). In practice one often chooses the time step
such that the acceptance rate becomes about 80%, which is on the safe side, but
still not too far from this instability limit.
It should be noted that the acceptance rate depends on the difference in the total
energy of the system before and after the trial step. The total energy is an extensive
quantity: it scales linearly with the volume. This implies that discrete time step errors
will increase with volume. To see how strong this increase is[19], we note that the
error in coordinates and momenta after many steps in the leap-frog/Verlet algorithm
is of orderh^2 per degree of freedom (see Problem A3). This is then the deviation
in the energy over the microcanonical trajectory, and we shall denote this deviation
HMD. The energy differences obtained including the acceptance/rejection step are
calledHMC; that is, if the trajectory is accepted,HMCis equal toHMD,but
if the step is rejected,HMC =0. IfHMDaveraged over all possible initial
configurations was to vanish, the acceptance rate would always be larger than
0.5, as we would have as many positive as negative energy differences (assuming
that the positive differences are on average not much smaller or larger than the
negative ones), and all steps with negative and some of the steps with positive energy
difference would be accepted. However, the net effect of the acceptance/rejection
step is tolowerthe energy, and since the energies measured with this step included
remain on average stationary,〈HMD〉must be positive. The fact that the energy
remains stationary implies that〈HMC〉=0 and this leads to an equation for