15.4 Algorithms for lattice field theories 487
The procedure can be implemented straightforwardly. The equations of motion
in the leap-frog form read
pn(t+h/ 2 )=pn(t−h/ 2 )+hFn(t); (15.66a)
φn(t+h)=φn(t)+hpn(t+h/ 2 ), (15.66b)where the forceFn(t)is given by
Fn(t)=∑
μ[φn+μ(t)]−( 2 d+m^2 )φn(t)− 2 gφ^3 n(t), (15.67)where
∑
μdenotes a sum overallneighbours. Refreshing the momenta should be
carried out with some care. We refresh the momenta at the time stepstfor which
the field valuesφnare evaluated in the leap-frog algorithm. However, we need the
momenta in the leap-frog algorithm at times precisely halfway between these times.
Therefore, after the momentum update, we must propagate the momenta over half
a time steph:
pn(t+h/ 2 )=pn(t)+hFn(t)/2, (15.68)
and then the integration can proceed again.
This method contains a tunable parameter: the refresh rate. It turns out that the
efficiency has a broad optimum as a function of the refresh rate[16]. Having around
50 steps between the all-momenta updates with a time steph=0.1 is quite efficient.
If we refresh after every time step, the system will essentially carry out a random
walk in phase space as the small steps made between two refreshings are nearly
linear, and the direction taken after each refreshment step is approximately random.
If we let the system follow its microcanonical trajectory for a longer time, it will
first go to a state which is relatively uncorrelated with respect to the previous one.
The momentum refreshings then ensure that the canonical distribution is satisfied;
however, the fact that the energy is not conserved, but may change by an amount (on
average) ofO(h^2 )during the MD trajectory, causes deviations from the canonical
distribution of the same order of magnitude.
This method is obviously more efficient than refreshing after each step, as the
distance covered by a random walker increases only as the square root of the num-
ber of steps made. If we wait too long between two refreshings, the simulation
samples only a few different energy surfaces which is not representative for the
canonical ensemble. The optimum refresh rate is therefore approximately equal to
the correlation time of the microcanonical system.
The Metropolis-improved MD methodThe leap-frog algorithm introduces systematic errors into the numerical simulation,
and the distribution will therefore not sample to the exact one. That is not necessarily
a bad thing: we can always write the distribution which is sampled by the MD