If the wavefunction of the system is
real and positive, as it happens in case of
ground state of a bose system, it can be treated as population
density distribution3.1
(3.26)
here is a positive constant, are coordinates of
a population element (so called walker) in -dimensional
configuration space,
gives the probability
to find a walker at time in vicinity of point .
Let us now interpret the action of the each of the three terms of
the Hamiltonian (3.15) on the population distribution
or, being the same, the action of the corresponding Green's
functions (3.21, 3.23,
3.25). In terms of Markov Chains the Green's
function is the
is the transition matrix
which determines the evolution of the distribution (see
eq.(3.14)).
The first term means diffusion of each of the walkers in
configuration space
(3.27)
here is a random value from a gaussian distribution
.
The second term describes the action of the drift force, which
guides the walkers to places in the configuration space, where the
trial wavefunction is maximal. This is the way how importance
sampling acts in this algorithm.
(3.28)
The corresponding Green's functions of these two steps
(3.21 - 3.23) are
normalized to one
. The normalization of
wavefunction is then conserved meaning that the number of
walkers remains constant.
The third term is the branching term
(3.29)
Here the corresponding Green's function (3.25)
is no longer normalized and, when the quantity in the exponent in
is negative (i.e. large values of local energy), then the density
of population decreases and vice-versa.
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