Average energy from partition function: Foundations

Before we can even read the formula $⟨ E ⟩ = - \partial_{β} ln Z$ , we must earn every piece of it. Below, each symbol gets three things: plain words, a picture, and why the topic needs it. They are ordered so each one leans only on the ones above it.

1. A microstate, and the label $i$

Picture. Think of a jar full of raffle tickets. Each ticket is one microstate. The index $i$ is just the number printed on the ticket — a name tag, nothing more.

Why the topic needs it. Everything we sum over is "all the microstates." Without a way to list them ( $i = 1, 2, \dots$ ) we cannot write a sum $\sum_{i}$ at all.

Figure — Average energy from partition function

2. Energy of a state, $E_{i}$

Picture. Draw a vertical "energy axis" (higher = more energy). Each microstate is a horizontal shelf at height $E_{i}$ . This is the energy-level diagram you see everywhere in this chapter.

Why the topic needs it. The whole question — "what is the average energy?" — only makes sense once each state carries a number $E_{i}$ . These numbers are the values we will average.

3. Temperature $T$ and the heat bath

Picture. A small cup of water floating in a giant lake. The lake's temperature never budges; the cup keeps swapping energy with it and settles to the lake's temperature.

Why the topic needs it. "In contact with a heat bath at temperature $T$ " is the whole setting. $T$ is the knob that decides whether low-energy states dominate (cold) or all states are roughly equal (hot).

4. Boltzmann's constant $k_{B}$

Picture. A currency exchange rate: kelvin on one side, joules on the other. Multiply temperature by $k_{B}$ and you get "the natural energy scale of jiggling at that temperature," namely $k_{B} T$ .

Why the topic needs it. Energies $E_{i}$ live in joules; temperatures live in kelvin. To compare them (as we must, inside an exponent) we need $k_{B}$ to make units match.

5. Inverse temperature $β$

Picture. A see-saw: as $T$ goes up, $β$ goes down, and vice versa. When $β$ is huge (cold), the exponent $- β E_{i}$ is a big negative number for high-energy states, crushing them.

Why the topic needs it. The final formula is $⟨ E ⟩ = - \partial_{β} ln Z$ — a derivative with respect to $β$ , not $T$ . Working in $β$ makes the algebra clean because $β$ multiplies $E_{i}$ directly in the exponent (no messy $1/ T$ buried inside).

6. The exponential $e^{- β E_{i}}$ — the Boltzmann weight

Now we combine the pieces. Why an exponential, of all functions?

Picture. A curve that starts high and decays. Low-energy shelves get tall bars; high-energy shelves get tiny bars. Cold ( $β$ large) makes the curve plunge fast; hot ( $β$ small) flattens it.

Why the topic needs it. These weights are the raw material of everything. The partition function is their sum; the probabilities are them normalized; the average energy is a derivative of their sum.

7. The partition function $Z$

Picture. Take every bar from the previous figure and stack them end to end into one long bar. That total length is $Z$ . As you warm the system, more bars grow, and $Z$ increases.

Why the topic needs it. Two jobs at once:

Normalizer — dividing each weight by $Z$ turns scores into genuine probabilities (they add to 1).
Generator — because $Z$ contains every $e^{- β E_{i}}$ , differentiating $Z$ reaches inside and pulls out the energies. This is the whole trick of the parent note.

8. Probability $p_{i}$ and the sum $\sum_{i}$

Picture. Slice a pie chart. Each slice is a microstate; the slice's angle is $p_{i}$ . Big Boltzmann weight → fat slice. The whole pie is $Z$ , so a slice's fraction of the pie is $w_{i} / Z = p_{i}$ .

Why the topic needs it. The average energy is $⟨ E ⟩ = \sum_{i} p_{i} E_{i}$ — literally "sum of value × probability." Both $\sum_{i}$ and $p_{i}$ must exist before that line means anything.

9. The average (expectation value) $⟨ E ⟩$

Picture. Back to the energy-level shelves: put a weight $p_{i}$ on each shelf and find the balance point of the whole stack. That balance height is $⟨ E ⟩$ . Cold → balance sits near the lowest shelf; hot → it rises toward the middle of the pack.

Why the topic needs it. This is the quantity the entire topic computes. The magic is that it equals $- \partial_{β} ln Z$ , so we never actually do the weighted sum by hand.

10. The derivative $\partial_{β}$ and the logarithm $ln$

These are the two tools the formula uses. State plainly why each tool and not another.

Why this tool? We need a factor of $E_{i}$ to appear next to each weight. Differentiating $e^{- β E_{i}}$ in $β$ manufactures exactly that factor: $\partial_{β} e^{- β E_{i}} = - E_{i} e^{- β E_{i}} .$ No other simple operation produces an $E_{i}$ out of thin air. That single fact is why a derivative — and specifically a derivative in $β$ — is the right instrument.

Why this tool? After differentiating we get $- \frac{1}{Z} \partial_{β} Z$ — a "divide by $Z$ , then differentiate" combo. The logarithm bundles that whole combo into one clean symbol $\partial_{β} ln Z$ . Bonus: $ln Z$ is exactly the object that gives the Helmholtz free energy $F = - k_{B} T ln Z$ , so using it links energy and free energy for free.

How the pieces assemble into the formula

Reading the figure above left-to-right: microstates → energies → weights → their sum $Z$ → its log → its slope in $β$ → flip the sign → average energy. Every arrow is one of the definitions above.

$§2 E_{i} \to §6 e^{- β E_{i}} \to §7 Z = i \sum e^{- β E_{i}} \to §10 ln Z \to §10 - \partial_{β} ln Z = ⟨ E ⟩ .$

Prerequisite map

The map shows two roads to the same destination: the honest sum road ( $p_{i} \to \sum p_{i} E_{i}$ ) and the generator road ( $ln Z \to - \partial_{β}$ ). The topic's whole point is that these two roads meet.

Where these foundations lead

The weights themselves are the Boltzmann distribution.
Their sum is the Partition function (canonical ensemble).
Differentiating $ln Z$ once → average energy (this topic); twice → Heat capacity and energy fluctuations.
Apply it to specific systems: Two-level system / Schottky anomaly, Quantum harmonic oscillator — thermal, and the Equipartition theorem.
The log itself is the free energy.

Return to the parent: 2.4.11 Average energy from the partition function.

Equipment checklist

Cover the right side and answer before revealing.

What is a microstate, in one sentence?

One completely specified possibility for the system (all positions, all spins fixed).

What does

E_{i}

stand for?

The energy the system has while sitting in microstate

i

What are the units of

k_{B}

and what does it do?

Joules per kelvin; it converts a temperature into an energy scale

k_{B} T

Write

β

in terms of

T

and say which way it points.

β = 1/ (k_{B} T)

; large when cold, small when hot.

Why must the state-weight be an exponential?

Only

e^{- β E}

multiplies correctly under adding energies, so relative likelihoods depend on energy gaps, not the zero of energy.

Write the partition function as a sum.

Z = \sum_{i} e^{- β E_{i}}

Give the two jobs

Z

performs.

It normalizes weights into probabilities, and it generates averages when differentiated.

Express the Boltzmann probability

p_{i}

p_{i} = e^{- β E_{i}} / Z

Define

⟨ E ⟩

as a weighted sum.

⟨ E ⟩ = \sum_{i} p_{i} E_{i}

What does

\partial_{β} e^{- β E_{i}}

equal, and why does it matter?

- E_{i} e^{- β E_{i}}

; it manufactures the factor

E_{i}

we need to average.

State the chain-rule identity that brings in

ln

\partial_{β} ln Z = \frac{1}{Z} \partial_{β} Z

Assemble the final formula for average energy.

⟨ E ⟩ = - \partial_{β} ln Z

Recall Fastest sanity check

If every $E_{i} \geq 0$ , must $⟨ E ⟩$ be positive? ::: Yes — the minus sign in $- \partial_{β} ln Z$ cancels the minus from the exponent, so a physically sensible positive average survives.