18 0.404 0.406 0.408 0.41 0.412 0.414 1 1.2 1.4 1.6 1.8 2 < N > L$_{t}$=320 L$_{t}$=540 L$_{t}$=640 L$_{t}$=960 0.4725 0.475 0.4775 0.48 0.4825 2 2.2 2.4 2.6 2.8 3 < N > L$_{t}$=320 L$_{t}$=540 L$_{t}$=640 L$_{t}$=960 0.532 0.534 0.536 0.538 3 3.2 3.4 3.6 3.8 4 < N > L$_{t}$=320 L$_{t}$=540 L$_{t}$=640 L$_{t}$=960 0.136 0.138 0.14 1 1.2 1.4 1.6 1.8 2 < N > L$_{t}$=320 L$_{t}$=540 L$_{t}$=640 L$_{t}$=960 0.214 0.216 0.218 2 2.2 2.4 2.6 2.8 3 < N > L$_{t}$=320 L$_{t}$=540 L$_{t}$=640 L$_{t}$=960 0.282 0.284 0.286 3 3.2 3.4 3.6 3.8 4 < N > L$_{t}$=320 L$_{t}$=540 L$_{t}$=640 L$_{t}$=960 1.864 1.866 1.868 1.87 1 1.2 1.4 1.6 1.8 2 < N > L$_{t}$=1000 L$_{t}$=1200 1.878 1.88 1.882 2 2.2 2.4 2.6 2.8 3 < N > L$_{t}$=1000 L$_{t}$=1200 1.888 1.89 1.892 3 3.2 3.4 3.6 3.8 4 < N > L$_{t}$=1000 L$_{t}$=1200 FIG. 5: The average particle number as a function of µ near the transitions between N and N + 1 for N = 1 , 2 , 3 at L = 8 and β = 0 . 43 (top), β = 0 . 5 (center) and β = 0 . 2 (bottom).The solid lines are fits to the data.