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C. Fluctuations 

To close this section we note an interesting finding on fluctuations. The variance of the density, as a function of α , with fixed β , exhibits a maximum at the discon- tinuous transition (see Fig. 11). For β values such that the transition is continuous, by contrast, no peak in var( ρ ) is observed at the transition. (Similar behavior is found, varying β with α fixed.) The large density fluctuations are associated with the presence of a shock separating high- and low-density regions, whose position fluctuates over the entire system. The position of maximum variance agrees to within uncertainty with the lines of the (discontinuous) phase transitions reported in Fig. 9. Analysis of the variance, however, appears to furnish less precise results than the method described above. The total energy exhibits fluctuations similar to those observed in the density, but we do not find any signal in var( J ) associated with the phase transitions. 

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