SCIENCE · INVESTIGATE AND EXPLAIN

Build a fixed-number partition sum with exclusion

Consider two identical spin-polarized noninteracting fermions and four nondegenerate one-particle states of energies 0,delta,2delta,3delta. Spin polarization means each listed state has capacity one. At fixed N=2, allowed configurations are unordered pairs of distinct indices: (0,1),(0,2),(0,3),(1,2),(1,3),(2,3). There are six, not sixteen.

Learning outcome: List all six configurations and normalize their weights for two temperature ratios. Reconstruct each level occupancy and total energy, then repeat with sixteen distinguishable configurations. Explain why a one-level occupancy greater than one would reject the spin-polarized fermion description.

Starting recap

Review finite sums, exponential and logarithmic functions, radial integration measures, first-order coupled equations and mass-energy units. Keep state counts, occupations, calibration records and unused tests distinct.

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Try

In a changed analytic packet set q=exp[−delta/(kBT)]=1/2. Pair weights sum to q+q²+2q³+q^4+q^5=35/32. Fermion ground-level occupancy is (q+q²+q³)/Z=4/5. The distinguishable comparison gives 2/(1+q+q²+q³)=16/15, illustrating the exclusion difference.

Open the experiment

Predict

What is fixed in both of these canonical comparisons?

Your prediction and reason

Experiment

List all six configurations and normalize their weights for two temperature ratios. Reconstruct each level occupancy and total energy, then repeat with sixteen distinguishable configurations. Explain why a one-level occupancy greater than one would reject the spin-polarized fermion description.

Keep fixed particle number while changing counting rules

Worked teaching model · synthetic values · Two identical spin-polarized noninteracting fermions occupy four nondegenerate one-particle states with energies 0, delta, 2delta, 3delta, canonical equilibrium at fixed N=2. Compare two distinguishable noninteracting classical particles in the same finite states, where repeated occupancy is allowed. This is not a grand-canonical approximation or a real-material equation of state.

One-level energy spacing: 1 eV Thermal energy kBT: 1 eV

Reset keeps your prediction, captures and practice history.

Same levels and temperature, different allowed configurations

Graph axes rescale independently; read units and values, not screen angles. 1: canonical spin-polarized fermions (solid).2: two distinguishable classical particles (dashed). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Same levels and temperature, different allowed configurations

canonical spin-polarized fermions

one-particle energy (eV)	mean occupancy
0	0.8807971
1	0.6944133
2	0.3055867
3	0.1192029

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

two distinguishable classical particles

one-particle energy (eV)	mean occupancy
0	1.287829
1	0.4737656
2	0.1742886
3	0.06411721

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

Calculated quantities under the declared measurement model
Quantity	Value	Unit
Canonical fermion configurations	6	integer
Distinguishable classical ordered configurations	16	integer
Fermion mean total energy	1.663195	eV
Classical mean total energy	1.014695	eV
Fermion ground-state occupancy	0.8807971	particles
Classical ground-state mean occupancy	1.287829	particles
Fermion total mean particle count	2	particles

Exclusion changes the allowed configurations before the thermal weights are normalized.

Enumerate six unordered distinct-level pairs i<j with Boltzmann weight exp[−(Ei+Ej)/kBT]. Normalize the pair partition function, then sum pair probabilities containing each level. For the distinguishable comparison there are 16 ordered pairs, including repetition, and mean occupancy2exp(−Ei/kBT)/z1. Both have exactly two particles; a chemical-potential Fermi function is not substituted for this small fixed-N canonical enumeration.

Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.

Exclusion changes the allowed configurations before the thermal weights are normalized.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Define the state and evidence

Derive the quantitative relation

In canonical equilibrium, a pair has weight exp[−(Ei+Ej)/(kBT)]. Sum these six weights to obtain Z, divide each by Z for its probability, and sum probabilities of pairs containing level i for its mean occupancy. Each occupancy lies between zero and one and their sum is exactly two. Mean energy is the sum of energy times occupancy.

Test a changed model or boundary

For a comparison with two distinguishable classical particles in the same four states, all sixteen ordered pairs are allowed, including both particles at one level. The one-particle sum is z1=sum exp(−Ei/kBT); mean occupancy is 2exp(−Ei/kBT)/z1. This can exceed one, without changing total particle number. The change is in allowed configurations, not a new energy spectrum.

Limit the inference

A grand-canonical Fermi function with an adjustable chemical potential is not substituted for this small fixed-number enumeration. At low temperature the fermions occupy the two lowest states, whereas the distinguishable comparison tends toward two particles in the lowest state. These are bounded statistical models, not proof of real-material equilibrium or an experimental spin preparation.

Worked original case

Investigation

Changed-case practice

Try the changed case before feedback

For q=1/2, find the fermion ground-level occupancy.

Numerical result in the requested units

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Sources and boundaries

Original derivations and virtual evidence checked 31 October 2026 against the primary explanation. All questions and numerical packets are original. Supplied theoretical distributions and synthetic records are not actual laboratory measurements. No nuclear, electrical, heating, beam or material-handling procedures are requested. Retain first reasoning, actual criterion critique, revision and a separately assigned unfamiliar task in the course evidence workspace; public keys do not authenticate mastery.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Which independent calibration, state-counting or physical-domain premise is needed, and which fresh observation could challenge the inference?

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

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Previous: Enumerate allowed labels before making a degeneracy claim Next: Keep a band-gap interpretation conditional on transport assumptions

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