Statistical mechanics
Connect microscopic states to temperature, entropy, fluctuations, phases, and quantum statistics.
Develop ensembles and partition functions, derive thermodynamic quantities, and learn when fluctuations or nonequilibrium resources can—and cannot—support useful work.
Before you begin
- • Calculus and probability
- • Level 2 thermodynamics
- • Introductory quantum mechanics
By the end, you can
- • Use microcanonical, canonical, and grand-canonical ensembles.
- • Derive observables from partition functions.
- • Calculate fluctuation scales and phase probabilities.
- • Separate equilibrium noise from nonequilibrium work resources.
Interactive model
Explore before calculating
Live laboratory
Canonical two-level ensemble
Change a level gap, temperature, and excited-state multiplicity. Boltzmann weights turn microscopic possibilities into measurable populations, energy, entropy, and heat capacity.
Excited population: 27.55%
Mean energy: 6.89 meV
Entropy: 0.589 kB
Heat capacity: 0.187 kB
These are equilibrium probabilities for a declared reservoir. Population fluctuations do not by themselves provide sustained cyclic work; a gradient, drive, measurement resource, or other nonequilibrium change must enter the ledger.
Level 3 · Undergraduate core teaching kit
Record the investigation. Teach the reasoning.
A learner-facing lab record and a course-specific instructor guide turn the live model into a repeatable classroom investigation.
Learner record
Two-level ensemble population and heat-capacity map
How do temperature, energy gap, and degeneracy turn microscopic possibilities into macroscopic averages?
Download learner recordInstructor guide
Teach for evidence, not button pushing
Learners derive qualitative ensemble behavior from Boltzmann weights and distinguish microstate multiplicity from energy alone.
Download instructor guideLesson 1 of 3
Microstates, macrostates, and ensembles
How can many microscopic configurations share one measured temperature and pressure?
A macrostate specifies coarse variables while many microstates realize them. Boltzmann entropy S=kB lnΩ counts compatible microstates.
An ensemble is a probability model over microstates given constraints. Choosing the ensemble means declaring what can exchange energy, particles, or volume.
Worked example
A macrostate has four times as many microstates as another. Find entropy difference.
- 1. Use ΔS=kB ln(Ω₂/Ω₁).
- 2. Insert ratio 4.
- 3. Keep logarithmic form or evaluate.
ΔS=kB ln4.
Try it
Multiplicity counting
Materials: Coins or binary-state simulation.
- 1. Enumerate heads counts for small N.
- 2. Compute binomial multiplicities.
- 3. Plot multiplicity versus macrostate.
- 4. Increase N.
Notice: Equilibrium-like macrostates dominate because vastly more microstates realize them.
Check your understanding: Does high entropy mean each microstate is disordered in a subjective visual sense?
Answer: No.
Entropy is defined from state probabilities or multiplicities under specified macroscopic constraints.
Lesson 2 of 3
Partition functions and thermodynamic derivatives
How can one weighted sum generate energy, entropy, heat capacity, and populations?
The canonical partition function Z=Σe⁻ᵝᴱ weights energy eigenstates in contact with a heat reservoir. Its logarithm generates thermodynamic quantities.
Free energy F=−kBT lnZ balances energy and entropy and determines equilibrium under fixed temperature and volume.
Worked example
For a two-level system with energies 0 and ε, find Z.
- 1. Weight the ground state by 1.
- 2. Weight excited state by e⁻ᵝε.
- 3. Add allowed-state weights.
Z=1+e⁻ᵝε and excited population is e⁻ᵝε/Z.
Try it
Two-level heat capacity
Materials: Notebook and temperature range.
- 1. Compute excited population versus T.
- 2. Compute mean energy.
- 3. Differentiate numerically for heat capacity.
- 4. Locate its maximum.
Notice: Heat capacity peaks where thermal energy competes with the level gap.
Check your understanding: Why is Z not usually a probability itself?
Answer: It is the normalization sum for Boltzmann weights.
Individual probabilities are e⁻ᵝᴱ/Z.
Lesson 3 of 3
Fluctuations, phases, and quantum statistics
When do collective fluctuations reveal a transition rather than a new energy reservoir?
Canonical energy variance is tied to heat capacity. Relative fluctuations usually shrink with system size but grow near some critical points.
Bosons and fermions obey different occupation statistics. Nonequilibrium gradients can drive work; equilibrium detailed balance forbids passive rectification into sustained cyclic output.
Worked example
If N independent contributions fluctuate with standard deviation proportional to √N, how does relative fluctuation scale?
- 1. Mean scales as N.
- 2. Standard deviation scales as √N.
- 3. Divide √N/N.
Relative fluctuations scale as 1/√N and become small macroscopically.
Try it
Ising-style phase exploration
Materials: Simple lattice simulation.
- 1. Run at high and low temperature.
- 2. Track magnetization and energy.
- 3. Sweep through transition.
- 4. Plot fluctuations and hysteresis checks.
Notice: Collective order and large fluctuations emerge from local interactions without implying free energy beyond the modeled bath.
Check your understanding: Can equilibrium thermal noise be passively rectified forever by an isothermal ratchet?
Answer: No.
The ratchet itself fluctuates; detailed balance cancels sustained directed work at one temperature.
Formula-to-meaning deck
Read the equation in ordinary language.
S=k_B lnΩ
Entropy measures multiplicity for equally likely constrained microstates.
Z=Σ_i e^(−βE_i)
Partition function normalizes canonical state weights.
F=−k_BT lnZ
Helmholtz free energy generates fixed-temperature equilibrium thermodynamics.
Independent practice
Problem set
Work each problem before opening its hint and solution.
1. Find the probability ratio p₂/p₁ for states separated by ΔE at temperature T.
Reveal hint
Divide their Boltzmann factors.
Reveal solution
p₂/p₁=e^(−ΔE/kBT).
2. For Z=1+e⁻ᵝε, find mean energy.
Reveal hint
Only the excited state contributes ε.
Reveal solution
⟨E⟩=εe⁻ᵝε/(1+e⁻ᵝε).
3. A system multiplicity doubles. Find ΔS.
Reveal hint
Use the multiplicity ratio.
Reveal solution
ΔS=kB ln2.
Derivation studio
Build the result, line by line.
Keep the assumptions visible so the mathematics remains auditable.
Starting point
Canonical distribution
Small system plus large isolated reservoir with fixed total energy
- 1. Probability of system energy E is proportional to reservoir multiplicity ΩR(Etot−E).
- 2. Expand reservoir entropy SR around Etot.
- 3. Use ∂S/∂E=1/T.
- 4. Exponentiate the first-order entropy change and normalize.
p_i=e^(−βE_i)/Z
Boltzmann weights arise from counting reservoir-compatible states.
Starting point
Energy variance and heat capacity
⟨E⟩=−∂lnZ/∂β
- 1. Differentiate ⟨E⟩ with respect to β.
- 2. Recognize second moment minus mean squared.
- 3. Use dβ/dT=−1/(kBT²).
- 4. Relate d⟨E⟩/dT to heat capacity.
Var(E)=kBT²C_V
Equilibrium energy fluctuations are quantitatively tied to response.
Computational notebook
Turn the model into an experiment.
Two-level and oscillator ensembles
How do discrete spectra shape energy, entropy, and heat capacity across temperature?
Inputs
- • Energy gap or oscillator frequency
- • Temperature grid
- • State cutoff for oscillator
Algorithm
- 1. Build Boltzmann weights.
- 2. Normalize probabilities.
- 3. Compute E, F, S, and C.
- 4. Test cutoff convergence.
Evidence to produce
- • Population and thermodynamic curves
- • Heat-capacity peak comparison
- • Numerical convergence report
Continue into the evidence
Source-linked next reading
Chapter 6: Getting Energy From the Vacuum
Thermodynamic and information boundaries for vacuum-energy protocols.
Lecture 8: Casimir effect
Equilibrium and driven boundary phenomena separated experimentally.
Chapter 5: Superfluid vacuum theory
Condensed-matter analogies, collective modes, and their limits.