Microstate Selection Probability Cross-Scan - Thermodynamics and Statistical Mechanics Step by Step

A microstate-selection-probability cross-scan varies the microstate across rows and reads off its selection probability, showing how the weight distributes over states. The arithmetic is exact only for the stated microstates, probabilities, and rows in the ideal ensemble model. Real systems add finite samples, preparation bias, correlations, non-equilibrium effects, rate-estimation error, and measurement/source uncertainty.

Three counted buckets show probability as selected count over the same total. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Bucket row 1 uses count over total

Bucket 0 has selected count 1 out of total 4, so the probability is 1/4.

P(\hbox{bucket }0)=\frac{1}{4}=\frac{1}{4}

Bucket row 2 uses count over total

Bucket 1 has selected count 2 out of total 4, so the probability is 1/2.

P(\hbox{bucket }1)=\frac{2}{4}=\frac{1}{2}

Bucket row 3 uses count over total

Bucket 2 has selected count 1 out of total 4, so the probability is 1/4.

P(\hbox{bucket }2)=\frac{1}{4}=\frac{1}{4}

Probability follows selected count over the same total

The total stays four. Counts one, two, and one give probabilities one fourth, one half, and one fourth.

\begin{array}{c|c|c|c}bucket&count&\Omega&P\\0&1&4&\frac{1}{4}\\1&2&4&\frac{1}{2}\\2&1&4&\frac{1}{4}\\\end{array}