My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 3.
QUANTUM MONTE CARLO METHODS
set of vectors {R i } corresponding to the points in configuration space visited by the
walker will be distributed according to the required probability density. To sample
a distribution P (R):
1. Start the walker at a random point R.
2. Propose a trial move to a point R ′ , chosen from some simple probability density
function T (R ′ ← R).
(
3. Accept the move with probability A(R ′ ← R) = min
4. Record the walker position, whether or not it has changed.
1, T (R←R′ )P (R ′ )
T (R ′ ←R)P (R)
)
.
5. If more points are required, return to step 2 and repeat.
Not all the points should be recorded: there is an equilibration period at the start
of the walk, during which the points generated depend on the initial position. In
addition, neighbouring points on the walk are usually correlated; this means that
several moves must be made for each independent sample.
Although in principle any reasonable 3 probability density function can be used
for T (R ′ ← R), the choice of function does affect the efficiency of the algorithm.
A function which proposes too many large moves produces too many rejections,
which is inefficient; in contrast, one which proposes only small moves takes longer to
generate uncorrelated points, which is also inefficient. In between these extremes is
a function which maximises the sampling efficiency; a commonly-used rule of thumb
is to aim for an acceptance rate of 50%.
To gain some insight into the way the algorithm works, consider a population of
walkers, with density n(R). Assume that the walkers have reached a steady state,
so that all information about starting positions has been lost; also assume that the
detailed balance condition is obeyed:
n(R)A(R ′ ← R)T (R ′ ← R) = n(R ′ )A(R ← R ′ )T (R ← R ′ ). (3.20)
3 T (R ′ ← R) must be ergodic; if T (R ′ ← R) is non-zero then T (R ← R ′ ) must also be.
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