Two-State Populations

Lower State Can Dominate the Population

In a two-state ensemble the lower-energy state carries the larger Boltzmann weight, so at low enough temperature its normalized probability can dominate the whole population. That dominance is exact only for stated weights, the two listed states, fixed temperature/model, and equilibrium sampling. Real systems add preparation bias, finite sampling, calibration, degeneracy, and uncertainty.

A lower state can dominate when its explicit weight is larger. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Weights set the population fractions

The lower state has three shares and the upper state has one share.

Plower=34,Pupper=14P_{\rm lower}=\frac{3}{4},\quad P_{\rm upper}=\frac{1}{4}
Two-state populationFractions are computed from explicit weights.energies=lower=0J|upper=2Jweights=lower=3|upper=1lowerFraction=3/4upperFraction=1/4upperLowerRatio=1/3weightOrigin=explicit integer weights

The upper fraction is one quarter

The upper-state population is one quarter of the finite ledger.

Pupper=14P_{\rm upper}=\frac{1}{4}
Upper population fractionThe energy labels alone do not set the fraction.energies=lower=0J|upper=2Jweights=lower=3|upper=1lowerFraction=3/4upperFraction=1/4upperLowerRatio=1/3weightOrigin=explicit integer weights