
In this section, the multilevel sampling procedure from sections 2.5.3 and 2.6.9 will be extended to the sampling with open paths for Bosons and distinguishable particles. Fermions will be discussed in section 5.2. A picture of an open path is shown in Fig. 5.1. We chose to put the open ends at time slice because we will apply the double reference method from section 2.6.4 to fermionic systems. For distinguishable particles, the open path is a single polymer that interacts with the other particles. The contributions to the action are calculated in the same way as is done for closed paths except for the diagonal pair action in the time slice containing the open ends. There, each open end contributes with the weight , which can be understood from Eq. 2.25.
Without interactions, the distribution of the open ends is by
definition given by the free particle density matrix in
Eq. 2.11. This equation will be used in the free
particle sampling method for the generation of new path sections that
contain the open ends. For closed paths, one samples the new positions
from a Gaussian distribution centered at the midpoint between the
slice above and below (Eq. 2.63) because of the two
spring terms in the free particle action. The open ends are only
connected in one direction in imaginary time. Therefore, the free
particle sampling distribution for open ends being connected to
or reads,