The primitive approximation (2.23) for the Green's function has a
first-order of accuracy, i.e. the resulting energy has a linear dependence on the
timestep. It means that measurements with different time-steps are necessary in
order to make extrapolation to zero timestep
as the dependence on
the timestep is very strong. Instead one can consider a higher order approximations.
One of the possible second-order expansions is
This expansion leads to a quadratic dependence of the energy on the timestep which makes this approximation very useful. The point is that each calculation has an intrinsic statistical error, which depends on the length of the calculation and on the variance of the measured quantity. Once the desired level of accuracy is chosen one can make a study of the dependence on the timestep and adjust it to the maximal value, which still gives an error smaller than the desired statistical error. Using this timestep one can avoid the extrapolation procedure at all.
Here is the summary of the higher order scheme used in the calculations. One step
which propagates the system in the imaginary time from to
.
The walker is moved from position
to position and is replicated
with the weight calculated during the branching
1) Gaussian jump (2.26):
2) Drift force (2.28):
3) Branching (2.30):