Two-State Population Family Cross-Scan - Statistical Physics: Ensembles, Weights, and Fluctuations Step by Step

Explicit two-state weights set fractions and upper-lower ratios. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Population row 1

Lower weight 3 and upper weight 1 make total 4. The upper fraction is 1/4 and upper over lower is 1/3.

P_{\rm upper}=\frac{1}{4},\quad \frac{N_{\rm upper}}{N_{\rm lower}}=\frac{1}{3}

Population row 2

Lower weight 5 and upper weight 2 make total 7. The upper fraction is 2/7 and upper over lower is 2/5.

P_{\rm upper}=\frac{2}{7},\quad \frac{N_{\rm upper}}{N_{\rm lower}}=\frac{2}{5}

Population row 3

Lower weight 1 and upper weight 1 make total 2. The upper fraction is 1/2 and upper over lower is 1.

P_{\rm upper}=\frac{1}{2},\quad \frac{N_{\rm upper}}{N_{\rm lower}}=1

Population fractions are explicit-weight fractions

The scan changes only the integer weights. The energy labels stay as training labels, so the population fractions follow the weight ledger.

\begin{array}{c|c|c|c|c}w_l&w_u&Z&P_u&N_u/N_l\\3&1&4&\frac{1}{4}&\frac{1}{3}\\5&2&7&\frac{2}{7}&\frac{2}{5}\\1&1&2&\frac{1}{2}&1\\\end{array}