Samples

From PIMC++

In path integral Monte Carlo (more generically Markov chain Monte Carlo using the Metropolis move), the probability of accepting a move is where is the change in action and is the probability of proposing a move to position b given that you've started in position a.

To understand why the bisection move with free particles (i.e. using only the kinetic action ) accepts 100% of the time, we should understand what is and what is. To do this, let's look at the simplest example of a bisection move.

Consider a single, free particle at three consecutive time slices along a path, , , and . In this move, we will keep and fixed, and will update the position of . We wish to do this optimally, so that it samples the right distribution given by

where is the free-particle density matrix given by

Writing this product explicitly, we have

Expanding the squares and grouping the -dependent terms, we have

where . We have separated out the -dependence and it is a simple gaussian centered at the midpoint of the line segment from to . This is probably the origin of the name "bisection move". This is special case of a more general result that the product of two Gaussian distributions is always another Gaussian distribution.

There exist simple methods to sample a point from a Gaussian distrubution. We use these methods, then, to randomly choose a value for according to above.

The new (old) action for this system would be (when we have only the kinetic action)

(the probability of selecting our new midpoint (the middle bead)) is which when simplified geometrically ends up being

When we take the ratio of the new to old, the term cancels out of the probability and we get the whole acceptance probability to be 1.

It is important to sample the kinetic action exactly even when a potential term is added because failure to do so will cause very bad acceptance ratios.