Two-State Population Family Cross-Scan - Thermodynamics and Statistical Mechanics Step by Step

A two-state-population-family cross-scan varies the temperature or energy gap across rows, showing how the two normalized populations shift as the weight ratio changes. The arithmetic is exact only for the stated energies, weights, populations, and rows in the ideal two-state ensemble model. Real systems add preparation bias, finite samples, degeneracy/model choices, thermalization limits, detector/counting calibration, finite precision, and measurement uncertainty.

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}