Figure 1.
Direct and indirect interactions.
For the five particle system shown here, with particle 1 as the selected particle, the direct interactions are shown as solid lines, and indirect interactions as dotted lines.
Figure 2.
For a 4 particle system, the set of all possible microstates are partitioned into subsets of states as follows. With particle 1 as the selected particles, the microstates are first partitioned into two subsets, one with particle 1 in state
and another with
, with contributions to the partition function corresponding to
and
respectively (Eq. (3)). Each of these subsets are further partitioned into subsets with the same number of particles,
, in state
, with contributions to the partition function corresponding to
and
in Eq. (10) and (11) respectively.
Figure 3.
Consider an 8-particle system, with particle 1 being the selected particle. First, the split-merge algorithm recursively separates the particles, other than the selected particle, into a hierarchical binary tree. Next, the elementary symmetric function (ESF) for the leaf nodes, which consist of a single particle, are calculated. The ESF for all the other nodes, are then computed by recursively, starting from the bottom, merging the ESF from the two branches for each node using Eq. (21).
Table 1.
Number of Monte Carlo (MC) steps.
Table 2.
Accuracy comparison.
Figure 4.
Accuracy as a function of system size.
Accuracy for the direct interaction algorithm (DIA) and the Metropolis Monte Carlo (MC) method as a function of system size (log-log scale). Accuracy is calculated as the RMS error relative to the exact value. The number of steps is chosen such that the computation time for the MC method is at least 10 times the computation time for the DIA (Table 1).
Figure 5.
Calculated values for the 128×128 2D Ising system.
(a) Thermal average magnetization , (b) internal energy
and (c) heat capacity
, as a function of dimensionless temperature
for the 128×128 2D Ising system, as calculated by the direct interaction algorithm (DIA), the Metropolis Monte Carlo (MC) method, and the exact solution. The number of steps for the MC method was chosen such that the MC computation time is at least 10 times the computation time for the DIA. Connecting lines are shown to guide the eye.