Statistical Mechanics — formula sheet
Every boxed result from the 15 derivations of this unit on one page, for revision. Each title links to the full derivation, with its assumptions and where the formula breaks down. Constants: physical constants table.
Boltzmann Entropy from Microstate Counting
Entropy is the logarithm of the number of equally likely microscopic configurations consistent with the macrostate, scaled by Boltzmann's constant. The logarithm is not decorative: it is the unique map that turns the multiplicative combinatorics of independent systems into the additive bookkeeping thermodynamics demands. Doubling the system size squares \(\Omega\) but only doubles \(S\).
Temperature, Pressure and Chemical Potential from Entropy Maximization
The entropy surface \( S(E,V,N) \) encodes all three intensive variables as its slopes. A steeper rise of entropy with energy means a colder body (larger \(1/T\)): adding a little energy buys a lot of extra multiplicity. Two bodies in contact equilibrate to a common slope in each direction, which is exactly \(T_A=T_B\), \(p_A=p_B\), \(\mu_A=\mu_B\). Off equilibrium, the second law \( dS_{\text{tot}}\ge 0 \) makes energy flow from high \(T\) to low \(T\), volume grow where \(p\) is higher, and particles move from high \(\mu\) to low \(\mu\).
The Partition Function and F = -kT ln Z
Everything thermodynamic is a derivative of ln Z. Compute the single scalar Z from the spectrum; its logarithm is (up to −kT) the free energy. From F(T,V,N) the standard thermodynamic derivatives deliver the rest: S=−(∂F/∂T)V,N, pressure P=−(∂F/∂V)T,N, chemical potential μ=(∂F/∂N)T,V, and U=F+TS. The differentiation with respect to β pulls down factors of Ei, which is why log-derivatives of Z generate energy moments (mean, variance, …).
Gibbs Entropy and the Maximum-Entropy Derivation of the Ensembles
Each ensemble is the maximum-entropy (least-biased) distribution given exactly the macroscopic information imposed on it: nothing but normalization gives the flat microcanonical weight; adding the mean energy tilts the weights into the Boltzmann exponential with slope \( \beta \); adding the mean particle number adds the fugacity factor \( e^{\beta\mu N_i} \). The Lagrange multipliers are not free parameters — they are the inverse temperature and chemical potential that tune the averages to their prescribed values.
Energy Fluctuations and Heat Capacity
The mean-square spread of a system's instantaneous energy about its average is set entirely by how much its average energy changes with temperature. A body that soaks up heat readily (large CV) is also a body whose energy visibly trembles; a body that barely responds to temperature is one whose energy is nearly frozen. Since ⟨(ΔE)²⟩ ≥ 0 always, the relation also proves CV ≥ 0: a stable system cannot have negative heat capacity.
The Equipartition Theorem
Every independent coordinate or momentum that enters the Hamiltonian as a pure square carries, on average, an energy \(\tfrac{1}{2}k_B T\) — and remarkably the value is universal: it does not depend on the stiffness \(\alpha\), the mass, or the frequency of the mode, only on temperature. A stiff spring and a floppy one store the same thermal energy; the stiff one simply achieves it with a smaller mean-square displacement, since \(\langle x^2\rangle = k_B T/\alpha\).
The Gibbs Paradox and Indistinguishability
The entropy per particle is set by the logarithm of the number of thermal cells available to one particle, \(V/N\) divided by the quantum volume \(\lambda^3\). The \(1/N!\) that produced the intensive argument \(V/N\) is the counting of indistinguishable permutations; it makes \(S\) extensive and quietly encodes that identical particles have no labels. The formula is trustworthy only while \(n\lambda^3\ll1\), i.e. while cells are sparsely occupied and Maxwell–Boltzmann statistics hold.
Ideal Gas Law and Sackur-Tetrode Entropy
The classical ideal-gas law and the absolute entropy of a monatomic gas fall out of one partition function. The entropy per particle is \(k_B\) times the logarithm of the number of thermal cells available to each particle, \(V/(N\lambda^3) = 1/(n\lambda^3)\), plus a fixed \(\tfrac{5}{2}\). The pressure knows nothing of \(h\) or \(m\); the entropy knows both, because counting states requires the quantum measure \(h^{-3}\) and the thermal wavelength shrinks with \(\sqrt{m}\).
Bose-Einstein and Fermi-Dirac Occupation Numbers
A single-particle level at energy \(\varepsilon\) is, on average, occupied by \(\langle n\rangle\) particles. When \(\varepsilon \gg \mu\) the exponential dominates and both statistics give a small, Boltzmann-like occupation. When \(\varepsilon \to \mu\) the denominators diverge in opposite ways: bosons pile up without bound (\(\langle n\rangle \to \infty\)), while fermions saturate at \(\langle n\rangle^{\text{FD}} \to 1/2\) exactly at \(\varepsilon=\mu\) and approach \(1\) below it — the Pauli ceiling. The chemical potential \(\mu\) is the energy at which one more particle costs no net free energy; it is fixed by requiring \(\sum_k \langle n_k\rangle = N\).
Planck's Law from the Photon Gas
The spectral energy density rises as \(\nu^2\) at low frequency (each new mode carries \(\sim k_BT\), the Rayleigh–Jeans regime) and is cut off exponentially once \(h\nu\gtrsim k_BT\), because populating a high-energy mode costs more than the thermal energy available. The peak sits at a frequency proportional to \(T\), and the area under the curve — the total energy — grows as \(T^4\). All three classical laws (Rayleigh–Jeans, Wien, Stefan–Boltzmann) are limits or moments of this one function.
Debye Model of Phonon Heat Capacity
The heat capacity of an insulating solid is a single universal function of \( T/\Theta_D \). At high temperature every one of the \( 3N \) modes is thermally excited and carries \( k_B \), giving Dulong–Petit. As \( T \) falls below \( \Theta_D \), modes with \( \hbar\omega > k_BT \) freeze out; because the surviving long-wavelength modes have a density of states \( \propto \omega^2 \), the number of active modes scales as \( T^3 \) and so does \( C_V \). The Debye temperature \( \Theta_D \) marks the crossover: stiff, light solids (diamond, \( \Theta_D\approx 2230\,\mathrm{K} \)) stay "quantum-frozen" to high \( T \); soft, heavy ones (lead, \( \Theta_D\approx 105\,\mathrm{K} \)) reach the classical plateau near room temperature.
Degenerate Fermi Gas and the Sommerfeld Expansion
The Fermi energy is set purely by density and mass — denser or lighter gases push electrons to higher momenta. Even at \(T=0\) the gas exerts a large pressure \(P_0=\tfrac25 n\varepsilon_F\) because the Pauli principle forbids the particles from all sitting at rest. As temperature rises, only the fraction \(\sim T/T_F\) of electrons within \(k_BT\) of the Fermi surface can absorb energy, so each contributes \(\sim k_B\) but only \(\sim(T/T_F)N\) of them participate, giving \(C_V\sim Nk_B(T/T_F)\) — linear, not constant, and tiny at room temperature.
Bose-Einstein Condensation
Condensation sets in when the phase-space density \(n\lambda_T^3\) reaches \(\zeta(3/2)\approx2.612\) — physically, when the thermal wavelength grows to the interparticle spacing so wavepackets overlap. Below \(T_c\) the excited states are "full" at their maximum thermal capacity \(\propto T^{3/2}\), and every particle beyond that capacity condenses into the ground state, giving a condensate fraction that rises from 0 at \(T_c\) to 1 at \(T=0\).
Mean-Field Ising Model and Spontaneous Magnetization
The magnetization per spin is determined self-consistently: the field each spin feels is set by the average alignment of its \(z\) neighbours. Above \(T_c\) thermal agitation wins and \(m=0\) is the only solution; below \(T_c\) two mirror-image ordered states \(\pm m\) appear and the system must pick one, breaking the up–down symmetry. The order parameter grows continuously from zero with a square-root law, so the transition is second order (continuous).
Landau Theory and Critical Exponents
Minimising a symmetry-constrained quartic free energy gives an order parameter that turns on as \((T_c-T)^{1/2}\), a susceptibility that diverges as \(|T-T_c|^{-1}\), a critical isotherm \(m\propto h^{1/3}\), and a specific-heat jump rather than a divergence. These are the mean-field exponents; they satisfy the scaling relations \(\alpha+2\beta+\gamma=2\) (Rushbrooke: \(0+1+1=2\)) and \(\gamma=\beta(\delta-1)\) (Widom: \(1=\tfrac12\cdot2\)) exactly, and \(\gamma=\nu(2-\eta)\) (Fisher) with \(1=\tfrac12\cdot2\). The hyperscaling relation \(2-\alpha=\nu d\) holds only at \(d=4\), the marker of the upper critical dimension.