Average energy from partition function: Question bank

Definition Symbols used on this page (define once, use freely)

$k_{B}$ = Boltzmann's constant, the fixed conversion factor between temperature and energy ( $k_{B} \approx 1.38 \times 1 0^{- 23} J/K$ ). It appears whenever temperature $T$ must be turned into an energy scale $k_{B} T$ .
$β \equiv \frac{1}{k _{B} T}$ = the inverse temperature. Large $β$ means cold, small $β$ means hot.
$Z$ = the partition function. For discrete microstates $i$ of energy $E_{i}$ , $Z = \sum_{i} e^{- β E_{i}}$ ; for a continuous spectrum, $Z = \int d E ρ (E) e^{- β E}$ . It is the normalizing sum of all Boltzmann weights.
$p_{i}$ = the Boltzmann probability of microstate $i$ , i.e. $p_{i} = \frac{e ^{- β E_{i}}}{Z}$ , the fraction of the time the system sits in state $i$ . By construction $\sum_{i} p_{i} = 1$ .
$ρ (E)$ = the density of states, i.e. $ρ (E) d E$ counts how many microstates have energy in the window $[E, E + d E]$ .
$ℏ \equiv \frac{h}{2 π}$ = the reduced Planck constant ( $h$ = Planck's constant). It sets the spacing of quantum energy levels, e.g. the oscillator's $ℏ ω$ .
$F$ = the Helmholtz free energy, defined by $F = - k_{B} T ln Z$ — the thermodynamic potential natural to a system at fixed temperature. See Helmholtz free energy F = -kT ln Z.
$C_{V}$ = the heat capacity at constant volume, $C_{V} = \frac{\partial ⟨ E ⟩}{\partial T}$ — how much energy the system absorbs per unit temperature rise. See Heat capacity and energy fluctuations.
Oscillator energy convention on this page: we measure the quantum harmonic oscillator's levels from its ground state, $E_{n} = n ℏ ω$ with $n = 0, 1, 2, \dots$ , so that $E_{0} = 0$ . (Adding the true zero-point $\frac{1}{2} ℏ ω$ only shifts $⟨ E ⟩$ by a constant — see the "adding a constant" trap below.)

The recurring theme: $⟨ E ⟩ = - \frac{\partial ln Z}{\partial β}$ where $β = 1/ k_{B} T$ , and every trap below hides in a sign, a $1/ Z$ , a missing Jacobian, or a limit.

True or false — justify

True or false:

⟨ E ⟩ = - \partial_{β} Z

is the average energy.

False — you must divide by

Z

. The bare

- \partial_{β} Z = \sum_{i} E_{i} e^{- β E_{i}}

is unnormalized; the correct object is

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

True or false:

⟨ E ⟩

is always positive.

False in general — it inherits the sign of the

E_{i}

. If you shift all energies down (e.g. levels

- ε, 0

⟨ E ⟩

can be negative; only when every

E_{i} \geq 0

⟨ E ⟩ \geq 0

True or false: adding a constant

E_{0}

to every energy level changes

⟨ E ⟩

E_{0}

True —

Z \to e^{- β E_{0}} Z

, so

ln Z \to ln Z - β E_{0}

and

- \partial_{β} ln Z \to ⟨ E ⟩ + E_{0}

. Energy has no absolute zero; only differences are physical.

True or false:

⟨ E ⟩ = k_{B} T^{2} \partial_{T} ln Z

contradicts

⟨ E ⟩ = - \partial_{β} ln Z

False — they are identical. Since

β = 1/ k_{B} T

, the chain rule gives

\partial_{β} = - k_{B} T^{2} \partial_{T}

, so the two forms are the same equation dressed in different variables.

True or false:

ln Z

is dimensionless.

True —

Z = \sum_{i} e^{- β E_{i}}

is a pure sum of dimensionless Boltzmann weights, so

ln Z

is a pure number. (For continuous

Z

one restores dimensionlessness with a phase-space factor like

h

True or false: for a two-level system

⟨ E ⟩

can exceed

ε

False — it interpolates between

0

(cold) and

ε /2

(hot), never reaching even

ε

. With only two states, the mean is a weighted average of

0

and

ε

that maxes out at

ε /2

when both are equally likely.

True or false: at

T \to \infty

every system's

⟨ E ⟩

diverges.

False — for a bounded spectrum (like the two-level system) it saturates at the arithmetic mean of the levels. Divergence at high

T

happens only for unbounded spectra (e.g. the oscillator).

True or false: the second derivative

\partial_{β}^{2} ln Z

can be negative.

False — it equals the variance

⟨ E^{2} ⟩ - ⟨ E ⟩^{2} = σ_{E}^{2} \geq 0

, so

ln Z

is convex in

β

. A negative value would mean an imaginary energy spread, which is impossible.

Spot the error

"Since energy rises with temperature,

⟨ E ⟩ = + \partial_{β} ln Z

Sign error.

\partial_{β} e^{- β E} = - E e^{- β E}

, so the minus is baked into the Boltzmann exponent:

⟨ E ⟩ = - \partial_{β} ln Z

. (And note raising

T

lowers

β

, so intuition about

T

shouldn't be applied to

β

signs.)

⟨ E ⟩ = k_{B} T \partial_{T} ln Z

because I converted

\partial_{β}

\partial_{T}

Missing a factor of

T

. The correct Jacobian is

\partial_{β} = - k_{B} T^{2} \partial_{T}

, giving

⟨ E ⟩ = k_{B} T^{2} \partial_{T} ln Z

— the power is

T^{2}

, not

T

"For the oscillator I'll write

Z = \sum_{n} e^{- β n ℏ ω} = \frac{1}{1 + e ^{- β ℏ ω}}

Wrong closed form. The geometric series

\sum_{n} r^{n} = \frac{1}{1 - r}

has a minus in the denominator:

Z = \frac{1}{1 - e ^{- β ℏ ω}}

. The

1 + \dots

form is the two-level partition function, a different problem.

C_{V} = \partial ⟨ E ⟩ / \partial β

Wrong variable. Heat capacity is defined per unit temperature:

C_{V} = \partial ⟨ E ⟩ / \partial T = k_{B} β^{2} σ_{E}^{2}

. Differentiating in

β

gives

- σ_{E}^{2}

, not

C_{V}

"Because

p_{i} = e^{- β E_{i}} / Z

, the highest-energy state is the most probable at high

T

Backwards emphasis. At high

T

(

β \to 0

) all weights

e^{- β E_{i}} \to 1

, so states become equally likely — no state dominates. Low-energy states dominate only at low

T

"Equipartition gives

\frac{1}{2} k_{B} T

per coordinate."

Imprecise. It gives

\frac{1}{2} k_{B} T

per quadratic term in the energy. A 1D oscillator has two (kinetic

\frac{p ^{2}}{2 m}

and potential

\frac{1}{2} k x^{2}

) →

k_{B} T

; a free particle's single coordinate contributes nothing without a quadratic potential.

Why questions

Why do we differentiate

ln Z

and not

Z

itself?

Because

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

by the chain rule, so the log folds the awkward "divide by

Z

, then differentiate" into one clean operation — and

ln Z

is exactly the free-energy object

F = - k_{B} T ln Z

Why does taking a derivative of

Z

"manufacture" the energy

E_{i}

Each term is

e^{- β E_{i}}

, and

\partial_{β}

of it brings down

- E_{i}

. So the derivative acts like an operator that inserts a factor of energy into every term of the sum — turning

\sum e^{- β E_{i}}

into

- \sum E_{i} e^{- β E_{i}}

Why is

Z

called a "generating function"?

Because successive

β

-derivatives generate successive moments:

- \partial_{β} ln Z = ⟨ E ⟩

\partial_{β}^{2} ln Z = σ_{E}^{2}

, and so on. One object encodes the entire statistical distribution of energy.

Why does the heat capacity measure energy fluctuations?

Because

C_{V} = k_{B} β^{2} σ_{E}^{2}

links the response (how

⟨ E ⟩

changes with

T

) to the spread

σ_{E}^{2}

. A system that stores heat readily is one whose energy fluctuates a lot — this is a fluctuation–dissipation relation.

Why must

\sum_{i} p_{i} = 1

, and where does that show up in

Z

Probabilities of mutually exclusive microstates must total one; dividing each weight by

Z = \sum_{i} e^{- β E_{i}}

enforces exactly this.

Z

is the normalizer, which is why it appears in the denominator of

p_{i}

Why does the classical (continuous) equipartition result not depend on the constant

a

E = \frac{1}{2} a x^{2}

Because

Z \propto β^{- 1/2}

with the

a

hiding only inside a

β

-independent constant. Since

⟨ E ⟩ = - \partial_{β} ln Z

ignores additive constants in

ln Z

, the stiffness

a

drops out entirely, leaving

\frac{1}{2} k_{B} T

Edge cases

T = 0

(

β \to \infty

), what is

⟨ E ⟩

for any system with a unique ground state?

It equals the ground-state energy

E_{0}

. All excited weights

e^{- β E_{i}}

die exponentially faster than the ground term, so the system freezes into its lowest state.

What happens to

Z

for the quantum oscillator as

T \to 0

e^{- β ℏ ω} \to 0

, so

Z \to \frac{1}{1 - 0} = 1

— only the

n = 0

ground state survives. Correspondingly

⟨ E ⟩ \to 0

, matching a frozen oscillator (with our convention

E_{0} = 0

What is

⟨ E ⟩

at high

T

for a two-level system, and why isn't it

ε

It is

ε /2

, the plain average of

0

and

ε

, because both states become equally probable. The mean of two equally likely values is their midpoint, never the larger one.

For an unbounded spectrum, why doesn't

Z

diverge at finite

T

The exponential suppression

e^{- β E_{i}}

beats the growth in the number/energy of states

ρ (E)

, so the sum (or integral

\int d E ρ (E) e^{- β E}

) converges for any

β > 0

. Convergence can fail only when

β \leq 0

What breaks if you try

β \to 0

exactly for the oscillator formula

⟨ E ⟩ = \frac{ℏ ω}{e ^{β ℏ ω} - 1}

The denominator

\to 0

, so the expression diverges — but taking the limit carefully (expand

e^{x} \approx 1 + x

) gives

⟨ E ⟩ \to k_{B} T

, the classical equipartition value. The apparent singularity is removable.

β \leq 0

(negative temperature) always just a mathematical pathology?

No — for a bounded spectrum,

β < 0

is a genuine physical state with inverted populations (more atoms in the upper level), realized in nuclear-spin and laser systems; it is "hotter than infinity." Only for an unbounded ladder does

β \leq 0

make

Z

diverge and become unphysical.

A system has a degenerate ground level (multiplicity

g

). Does

⟨ E ⟩

T = 0

change?

No —

⟨ E ⟩ \to E_{0}

still, because all

g

degenerate states share the same energy. Degeneracy alters

Z

's prefactor and the entropy, but not the mean energy at

T = 0

⟨ E ⟩ = - \partial_{β} ln Z

valid for a continuous spectrum where

Z

is an integral?

Yes — the derivation only used linearity of the derivative and the structure

e^{- β E}

, which survives replacing

\sum_{i}

\int d E ρ (E)

. The equipartition example is exactly this case.

Recall One-line summary of the whole trap set

Almost every trap is one of four sins: a dropped minus sign, a forgotten $1/ Z$ , a missing $k_{B} T^{2}$ Jacobian, or an illegal limit on a bounded-vs-unbounded spectrum. Guard those four and you're safe.

Related: Boltzmann distribution · Helmholtz free energy F = -kT ln Z · Heat capacity and energy fluctuations · Equipartition theorem · Two-level system / Schottky anomaly · Quantum harmonic oscillator — thermal