(a) Sketch of the unit vectors of the lattice. In (b) and (c) two representations of a unit cell with , are reported: (b) three-dimensional sketch and (c) string representation. In (b) gray numbered cubes represent exit sites, while the sphere represents the ensemble of inner sites. Assigned numbers establish the correspondence between exit sites in the three-dimensional (3D) sketch (gray cubes) and in the string representation (gray squares). The topology of exit sites in one cell is the same as the topology of the cells in the entire system; therefore it must be specified. The inner sites are distinguishable, but their spatial arrangement is ignored (this is why we employ a sphere for their 3D representation and white numbered squares in the string). In (d) two connected cells, named and , are represented. As can be seen, site 2 of cell communicates with site 5 of cell . The two cells can exchange particles using only these two connected sites. Since each site cannot host more than one particle, two adjacent cells can exchange at most one particle at time. This represents a constraint on the particle traffic.
The randomization (two-dimensional representation). During this operation, each cell is treated as a closed, independent canonical system. The input configuration (left) is mapped onto an output one (right) following an equilibrium criterion (see text for details). The dots represent particles in the exit sites, and the numbers represent particles in the inner sites. The total number of particles in each cell is given by the sum of the particles in the exit and inner sites. From the figure it is easy to verify that during randomization the number of particles in each cell is invariant.
The propagation (two-dimensional representation). During this operation, each pair of adjacent exit sites is treated as a closed, independent canonical system. In it, if only one exit site is occupied, then the occupying particle will jump into the other exit site with probability , here assumed as 1. If both exit sites are empty or occupied, then nothing occurs.
The occupancy distributions (straight lines) computed from Eq. (10) for loadings at various temperatures are shown in comparison with the hypergeometric distributions (dashed lines).
Conditional probability for at various temperatures. For low temperatures the particles tend to occupy preferably the inner sites, so is significant only for the lowest possible values of . Therefore the accessibility of the exit sites is low. As increases, accessibility of exit sites also increases and higher values of become possible.
Reduced variance as a function of loading at various temperatures.
Partial loading of the subsystem of exit sites (solid line) and partial loading (dashed line) of the subsystem of inner sites plotted together with respect to the total loading at various temperatures. For very high temperatures they increase in the same way because the two subsystems are equivalent. For low temperatures the two subsystems become very different; therefore the partial loadings increase with in different ways.
The chemical potential and (in the insets) the reduced variance for (top) and (bottom). The simulations were performed on a grid of cells for running times of time steps (black squares) and of time steps (white circles). Solid lines are fits through Eq. (8). is in units of kJ/mol.
The energy distributions around the phase transition (, ) for a system with cells. The distribution was computed averaging over time steps in (a) and over time steps in (b) (see text for further details).
The molar specific heat per cell for (top) and (bottom). Black squares and white circles indicate simulations and time steps long, respectively.
The variance of the occupancy distributions at the two limiting temperatures and , and at the intermediate temperatures of and .
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