Counting with Identical Particles

Teach me

Arrangement gallery · distinguishable (labels on)

Guided Inquiry

1 · The gallery of every way

Let’s start with the gallery. Three labelled particles, three levels. Each tile is one complete arrangement, drawn as a mini-ladder. Hover over or click a tile to load it on the big ladder — the letters A, B, C are the labels.

2 · Peel the labels off

Now here’s the key idea. Suppose swapping two particles no longer makes a different arrangement. What do you think — would the gallery then hold more arrangements, fewer, or exactly the same number?

3 · Throw the switch

Let’s see it happen. Press bosons on the statistics switch and watch the merge: tiles fly together, and each survivor gains a badge saying how many labelled tiles merged into it. What does the arrangements counter do?

4 · All three on one rung

Here’s a subtler question. Every arrangement in each gallery is equally likely. Compared with the labelled gallery, is finding all three particles on ONE level more likely for bosons, less likely, or exactly as likely? Have a think before you answer.

5 · Read it off the badges

You can check it yourself — the all in one level counter is now on screen. Switch between distinguishable and bosons and compare it. The crowded tiles carry badge ×1, the one-per-rung tile ×6: the merge devoured the spread-out arrangements.

6 · Forbid sharing

Now let’s add a new rule. Suppose no level may hold more than one of the three particles. How many of the ten boson arrangements do you think would survive that rule?

7 · The exclusion wall

Let’s test it. Press fermions and read both counters as tiles grey and fall. Then drag particles to 4, keeping 3 levels, and see what the gallery says.

8 · Statistics without forces

Notice there’s no force anywhere — only the rule for “different” changed. So put it to work: which rule you’ve just thrown explains why electrons stack into atomic shells instead of all falling to the lowest level?

Listen

Statistics — what counts as different

labels on: A, B, C, D name the particles

particles n3

levels g3

4 particles and 4 levels together would need 256 tiles — the gallery never exceeds 81

Results

▾

distinguishable — labels on

Arrangements

All in one level

3/27 = 1/9 = 0.111

No energy weighting — every arrangement is equally likely.

Distinguishable (Maxwell–Boltzmann counting): each labelled particle chooses a level independently.

Bosons: labels off, sharing allowed — stars and bars over occupancy vectors (nᵢ = number on level i).

Fermions: labels off and every nᵢ ≤ 1; identically zero when n > g — Pauli exclusion, as a count.

Badge on each boson tile: how many labelled tiles merged into it. The badges sum to gⁿ exactly.

All n particles on one level: g favourable tiles out of the gallery. Uniform ensemble, no energy weighting.

The Physics

Counting identical particles · what this sim shows, the physics behind it, and every control

At a glance

This sim lists every way to put n particles on g levels, then changes the rule for when two arrangements count as the same. The three rules give different counts and different odds, and they are the starting point for Bose–Einstein and Fermi–Dirac statistics.

What you'll learn

How the number of arrangements depends on n, g and the counting rule.
What a ×m badge records, and how it links one count to another.
How the chance of finding every particle on one level depends on the rule.
How Pauli exclusion shows up as a count, and what this sim leaves out.

One arrangement of 3 particles on 3 levels, drawn with labels (left) and without (right). Level 0 is the bottom rung.

What's on screen

Gallery tile: one arrangement, drawn as a mini-ladder. The gallery holds every arrangement allowed by the current rule.
Big ladder (left): the tile you hover or click, drawn larger.
Lettered discs: particles A, B, C, D, with labels on.
Plain discs: the same particles with the labels off.
×m badge: the tag on a tile when the labels are off.
Results: the Arrangements count and the All in one level share.

See itOpen the gallery for 3 particles on 3 levels, labels on. Watch for: a grid of mini-ladders, one per arrangement, with the big ladder on the left.

The controls

Press ⌖ on a row to see where that control sits in the sim.

Control	What it changes	Range
distinguishable (Statistics)	Labels on. The letters name the particles, and every assignment of levels is its own tile.	opens here
bosons (Statistics)	Labels off, sharing allowed. Swapping two particles gives the same tile, and each tile carries a ×m badge.	switch
fermions (Statistics)	Labels off, and no level may hold two. Tiles that break the rule are greyed out, and the badges dim.	switch
particles n	How many particles are placed. The gallery rebuilds.	2–4 opens at 3
levels g	How many levels there are. The gallery rebuilds.	2–4 opens at 3
Gallery tiles	Hover or click a tile to load it onto the big ladder.	click or hover
Results	Arrangements: how many tiles the current rule gives. All in one level: how many of them have every particle on a single level, as a fraction.	panel
↻ Reset	Returns n and g to 3 and 3.	top bar

Fixed: the levels carry no energies, so every tile in a gallery is equally likely. The gallery never holds more than 81 tiles, so n = 4 with g = 4 is blocked and the slider you moved snaps back.

Labelled particles give gⁿ arrangements

An arrangement is one complete way to put n particles on g levels. With labels on, each particle picks a level independently of the others, so there are gⁿ arrangements. The gallery draws every one of them; nothing is sampled.

The distinguishable count, the Arrangements readout with labels on. No division by n! anywhere.

The levels carry no energy, so nothing favours one tile over another: every tile in a gallery is equally likely. That is the only probability assumption the sim makes, and every share it reports is a count of tiles divided by a count of tiles.

See itRead the labelled count for 3 particles on 3 levels. Watch for: Arrangements reading 27 = 3³, one tile for each way to give A, B and C a level.

Taking the labels off merges tiles

Without labels, a tile is only its occupancy vector (n₀, n₁, …, n_g−1): how many particles sit on each level. Tiles that differ only by swapping particles become the same tile. Identical particles that may share a level are bosons, and counting the occupancy vectors gives the boson count:

Stars and bars: n particles in g levels is n + g − 1 symbols, n of them particles.

The badge on a tile is the number of labelled tiles that merged into it:

The badges always add up to gⁿ, because every labelled tile lands on exactly one boson tile.

Where the 27 labelled tiles of n = 3, g = 3 go.

See itSwitch from distinguishable to bosons for 3 particles on 3 levels. Watch for: the tiles flying together, each landing tile gaining a ×m badge, and Arrangements counting down to its boson value.

The crowded share depends on the rule

Take the g tiles with all n particles on one level. Each has m = 1, so none of them merges with another. Spread-out tiles merge many to one, up to n! when every particle sits alone on a level. The same g tiles are then compared with a different total:

The All in one level readout. P_FD = 0 is explained in the next section.

n, g	labelled	bosons
3, 3	3/27 = 0.111	3/10 = 0.300
2, 4	4/16 = 0.250	4/10 = 0.400

For every (n, g) the sim allows, the boson share is the larger. No force acts anywhere in the sim: the counting rule alone changes the odds.

See itRun the same switch, labelled to bosons, and read the share. Watch for: the labelled share 3/27 = 1/9 = 0.111 first, then the boson share 3/10 = 0.300 once the tiles have merged.

Exclusion removes every shared level

Identical particles that may not share a level are fermions. Of the boson tiles, only those with every nᵢ ≤ 1 survive:

The same occupancy vector, read under two rules.

Choose which n of the g levels are occupied. The count is zero when n > g.

A count of zero is a result, not an error: the sim shows 0 arrangements when the particles do not fit one per level. Exclusion also forbids putting two or more particles on one level, so for fermions the All in one level readout says "0 — never".

n g

These sliders move the sim too.

See itSwitch from bosons to fermions for 3 particles on 3 levels. Watch for: tiles with a shared level greying out and dropping, and Arrangements counting down to 1.

See itDo the same with 4 particles on 3 levels. Watch for: every tile dropping, and the gallery reading "0 arrangements".

Where it matters

The two rules for n = 3 on g = 4. Here a "level" is one single-particle state.

Atoms. Electrons are fermions, so no two share a single-particle state. They fill states from the bottom up, which gives atoms their shell structure. A real energy level can hold several electrons because it contains several states, such as the two spin states of an orbital.
Lasers and condensates. Photons are bosons. A laser puts a very large number of them into one mode, and in a Bose–Einstein condensate of ultracold atoms a large share of the atoms sit in the lowest state.
What the sim leaves out. It has no energies or temperature, so it does not produce the Bose–Einstein or Fermi–Dirac distributions; those need level energies on top of these counts. It stops at n ≤ 4 and g ≤ 4 because it draws every tile.

Go deeper: when levels far outnumber particles

All three counts approach gⁿ/n! only when g is much larger than n.

For g ≫ n almost every arrangement has the particles on different levels, so a boson tile or a fermion tile stands for about n! labelled tiles. Dividing gⁿ by n! is the "correct Boltzmann counting" used for a classical gas.
The sim is far from that limit. At n = 2, g = 4: W_BE = 10, gⁿ/n! = 8, W_FD = 6. At n = 3, g = 3: gⁿ/n! = 4.5, while W_BE = 10.

Summary

In one line. The rules differ only in what counts as the same arrangement: labelled particles are all different, bosons are the same under a swap, and fermions are also barred from sharing a level.

Key points

Labelled: gⁿ arrangements. Bosons: C(n+g−1, n). Fermions: C(g, n), which is 0 when n > g.
A ×m badge counts the labelled tiles merged into one boson tile, and the badges add up to gⁿ.
The g crowded tiles never merge, so their share is g/gⁿ labelled and g/C(n+g−1, n) as bosons.

Common mix-ups

"Bosons crowd because they attract, and fermions avoid each other because they repel." This sim has no interaction at all. The rule for what counts as different does all the work.
"The gallery shrank, so crowding got rarer." A smaller list is not a smaller share: the crowded tiles all survive, and the spread-out ones merge away.
"Identical particles can never share a level." That is the fermion rule only. Bosons share freely.

Things to try on your own

In bosons mode with n = 2, step g from 2 to 4 and note the All in one level share at each g.
For n = 3, find the smallest g at which the fermion gallery is not empty, then check the count against C(g, 3).
At n = 3, g = 3 in bosons mode, find the tile with badge ×6. Switch to distinguishable and count the tiles that also have one particle on each level.

Predict before reveal	Three gated predictions; the all-in-one-level counter appears only after the crux prediction is committed.
Cognitive conflict	The gallery shrinks 27 → 10 while crowding rises 1/9 → 3/10 — the shrinking-gallery intuition is named and rebutted in feedback.
Invariant made visible	Badges (×1, ×3, ×6) always sum to gⁿ: the boson tiles partition the labelled tiles exactly.
Operational numbers	Both counters are exact integers and exact fractions with decimals, computed from the same enumeration the gallery draws.
Empty is data	4 fermions on 3 levels renders “0 arrangements”, never an error — exclusion counted out.
Colour is redundant	Letters A–D carry particle identity; hue merely repeats it, and plain discs mark the labels being gone.