490 Computational methods for lattice field theories
〈HMD〉:
〈HMC〉= 0 =∑
{,P}Pacc∑(HMD)HMD
{,P}. (15.73)
UsingPacc=min[1, exp(−HMD)], and expanding the exponent, we find
0 =〈HMD〉−〈θ(HMD)(HMD)^2 〉 (15.74)where the theta-function restrictsHMDto be positive:θ(x)=0 forx<0 and 1
forx>0. We see that〈HMD〉is indeed positive and we furthermore conclude
that〈HMD〉=O(h^4 V)for of the order ofVdegrees of freedom. For the average
acceptance value we then find
〈Pacc〉=〈min(1, e−HMD)〉≈e−〈HMD〉=e−αh(^4) V
(15.75)
whereαis of order one. Therefore, in order to keep the acceptance rate constant
when increasing the volume, we must decreasehaccording toV−^1 /^4 , which implies
a very favourable scaling.
The Langevin method
Refreshing the momenta after every MD step leads to a Langevin-type algorithm.
Langevin algorithms have been discussed in Sections 8.8 and 12.2.4. In Section 8.8
we applied a Gaussian random force at each time step. In the present case we assign
Gaussian random values to the momenta at each time step as in Section 12.2.4. In
that case the two steps of the leap-frog algorithm can be merged into one, leading
to the algorithm:
φn(t+h)=φn(t)+
h^2
2
Fn(t)+hRn(t). (15.76)
The random numbersRnare drawn from a Gaussian distribution with a width of
1 – it is a Gaussian momentum, not a force (hence the pre-factorhinstead ofh^2 ).
Comparing the present approach with the Fokker–Planck equation discussed in
Section 12. 2. 4 , we see that when we takeρof the Fokker–Planck equation (12.42)
equal to exp(−S[φ]), Eq. (12.46) reduces to (15.76) if we putt = h^2. This
then shows immediately that the Langevin algorithm guarantees sampling of the
configurations weighted according to the Boltzmann distribution.
An advantage of this algorithm is the memory saving resulting from the momenta
not being required in this algorithm but, as explained in the previous section, the
method is not very efficient because the system performs a random walk through
phase space. The reason we treat this method as a separate one here is that there
exists an improved version of it which is quite efficient[ 17 ].We shall discuss this
algorithm inSection 15.5.5.