Thermodynamics & Statistical Mechanics
CollegeFrom microstates to macroscopic law
Chapter 14 treated temperature and entropy phenomenologically. Statistical mechanics derives them from counting. A macroscopic state (fixed energy, volume, particle number) corresponds to an enormous number of possible microscopic arrangements, or microstates, Ω. Boltzmann's foundational relation defines entropy directly from that count:
For a system in thermal contact with a much larger reservoir at temperature T, the probability of finding it in a microstate of energy Ei follows the Boltzmann distribution Pi ∝ e−Eᵢ/k_BT. Normalizing introduces the central object of the theory, the partition function:
Every thermodynamic quantity follows from Z: mean energy ⟨E⟩ = −∂lnZ/∂β (with β = 1/kBT), Helmholtz free energy F = −kBT lnZ, and entropy S = (⟨E⟩ − F)/T.
Why the second law is a matter of overwhelming probability
Chapter 14 called entropy "the deepest asymmetry in physics" and noted disordered states vastly outnumber ordered ones. Statistical mechanics makes this exact: entropy S = kBlnΩ increases in a spontaneous process simply because the system evolves toward whichever macrostate has overwhelmingly more microstates. A gas doesn't spread to fill a room because of some active "force of disorder" — it spreads because there are astronomically more ways for its molecules to be spread throughout the room than crammed in one corner, and random molecular motion will, essentially with certainty, sample the far more numerous "spread out" arrangements. The second law is not fundamental dynamics — Newton's (or Schrödinger's) equations are reversible — it is a statement about probability at the scale of 10²³ particles, where "overwhelmingly likely" is functionally indistinguishable from "certain".
pV = NkBT — Chapter 14's ideal gas law falls straight out of counting microstates, with no separate postulate needed.Worked example — the two-level system
Consider N independent particles, each with two possible energy states: 0 and ε. Find the partition function, mean energy, and high/low temperature limits.
- Single-particle partition function:
z = e−0/k_BT + e−ε/k_BT = 1 + e−ε/k_BT. - For N independent, distinguishable particles:
Z = zN. - Mean energy of one particle:
⟨E⟩ = −∂lnz/∂β = ε e−βε/(1+e−βε), withβ=1/kBT. Total:⟨Etot⟩ = N⟨E⟩. - Low-T limit (
kBT ≪ ε):e−βε → 0, so⟨E⟩ → 0— nearly every particle sits in the ground state, as expected near absolute zero. - High-T limit (
kBT ≫ ε):e−βε → 1, so⟨E⟩ → ε/2— particles are equally likely in either state, the maximum-entropy configuration.
This toy model is the backbone of paramagnetism (spins up/down in a field) and of Einstein's early model of specific heat in solids.
Why is entropy additive for two independent subsystems, given S = k_B ln Ω?
If subsystem A has ΩA microstates and subsystem B independently has ΩB, the combined system has ΩAΩB microstates (every combination). Then S = kBln(ΩAΩB) = kBlnΩA + kBlnΩB = SA + SB — the logarithm converts multiplicative counting into additive entropy.
What is the relationship between the partition function Z and the Helmholtz free energy F?
F = −kBT lnZ. Since Z encodes every accessible microstate weighted by its Boltzmann factor, its logarithm directly yields the free energy, from which pressure, entropy, and chemical potential all follow by differentiation.
Why does the two-level system's mean energy approach ε/2, not ε, at high temperature?
At high T both states become equally probable (probability ½ each), so the average energy is the simple average of the two levels: (0 + ε)/2 = ε/2. It can never exceed this because no state has energy above ε.
How does this chapter's statistical view resolve the paradox that Newton's laws are time-reversible but the second law is not?
Reversing every particle's velocity in a gas is a perfectly valid solution of the (reversible) microscopic equations — but it corresponds to one absurdly special microstate among the astronomical number available. Generic initial conditions overwhelmingly evolve toward higher-Ω macrostates. Irreversibility is a property of typical, macroscopic observation, not a violation of the reversible microscopic laws.
How the ideas connect
Every key idea in this chapter, branching from the core concept — use it to see the whole picture at a glance.
The key facts, visualised
Worked problems, step by step
Follow each solution line by line, then try to reproduce it on paper before moving on.
Example 1A two-level system has energies 0 and E. Write the partition function Z.
- Sum over both states with Boltzmann factors
- Z = exp(0) + exp(-E/kT)
Example 2For that two-level system, find the probability of the upper state.
- P(upper) = exp(-E/kT) / Z
- Z = 1 + exp(-E/kT)
Now you try
Work each one out first, then tap to reveal the worked answer.