Lecture 24–25 — The Canonical Ensemble and the Classical Ideal Gas#
Source: NPTEL Classical Physics, Mod-01 Lec-24 (The canonical ensemble) and Lec-25 (Connection between statistical mechanics and thermodynamics), Prof. V. Balakrishnan.
Lecture 22–23 built the entire structure of thermodynamics from one postulate applied to an isolated system’s own energy shell. But most real systems of interest — a single atom, a small protein, a cup of coffee — are not isolated at all; they sit inside a much larger environment that can freely exchange energy with them. This lecture asks what the microcanonical postulate implies for a small subsystem embedded in a huge heat bath, and the answer is the single most-used formula in all of statistical mechanics: the Boltzmann factor \(e^{-\beta\varepsilon}\).
The microcanonical distribution is a sharp Gaussian#
First, a sanity check on Lecture 22–23’s claim that fluctuations are negligible for macroscopic systems. Expanding \(\ln P(E)\) in a Taylor series about its maximum \(\bar E\) — the first-derivative term vanishes there by definition, and the second-derivative coefficient is negative since \(\bar E\) is a maximum — gives
a Gaussian, sharply peaked, with a variance proportional to the mean — the same Poisson-like scaling met twice already in Lecture 20–21. Since \(\bar E\) scales with the number of degrees of freedom, the relative width \(\sigma/\bar E \sim 1/\sqrt{\bar E}\) is utterly negligible for \(N\sim10^{24}\): this is precisely why thermodynamics can call \(\bar E\) the internal energy \(U\) and never mention fluctuations at all. But this argument used the fact that both subsystems \(A\) and \(B\) have huge numbers of degrees of freedom — exactly the assumption about to be dropped.
Shrinking the subsystem: the Boltzmann factor, derived#
Let \(A\) now be a small system — a handful of degrees of freedom — immersed in a heat bath \(B\) with far more, so that \(A\)’s energy \(\varepsilon\) satisfies \(\varepsilon \ll E_{\rm total}\). The probability that \(A\) has energy \(\varepsilon\) is still governed by the fundamental postulate applied to the whole isolated system, but now the roles are lopsided: \(A\)’s own \(\Omega(\varepsilon)\) plays no part in fixing \(P(\varepsilon)\), because it is entirely \(B\)’s response that matters,
Since \(\varepsilon \ll E_{\rm total}\), Taylor-expand \(\ln\Omega'(E_{\rm total}-\varepsilon)\) to first order in \(\varepsilon\) about \(E_{\rm total}\) — higher orders are negligible precisely because \(B\) is huge:
using exactly the definition of inverse temperature from Lecture 22–23, \(\beta \equiv \partial\ln\Omega'/\partial E'\), now evaluated for the bath. Exponentiating,
This is the Boltzmann factor, and the derivation is worth pausing on: it is not a separate postulate bolted onto the microcanonical one, but a direct consequence of it in the limit of a small system coupled to a large bath. All of \(B\)’s overwhelming internal complexity — its own energy levels, its own dynamics — collapses into exactly one number, its temperature. \(A\) itself is not in a definite energy state; it is constantly being driven through a whole distribution of states by the bath, and — for a quantum system — occupies not even a superposition of energy eigenstates but a genuine statistical mixture.
The partition function#
Normalizing \(P(\varepsilon_j) = e^{-\beta\varepsilon_j}/Z\) over every accessible state \(j\) (not merely every level — degenerate states must each be counted) defines the canonical partition function,
the continuum form using the density of states \(\rho(\varepsilon)\) derived in Lecture 22–23. \(Z\) is exactly a Laplace transform of \(\rho(\varepsilon)\) — a moment-generating function with \(\beta\) playing the role of transform variable — and this is the split that makes the whole formalism work: \(Z\) carries all the system’s own microscopic dynamics through \(\rho(\varepsilon)\), while the weight \(e^{-\beta\varepsilon}\) carries only the bath’s temperature. Different physical systems differ only through \(\rho(\varepsilon)\); the exponential factor is universal.
Average energy and the variance — \(C_V\) derived, not assumed#
Differentiating \(\ln Z\) with respect to \(\beta\) gives the average (internal) energy directly:
A second derivative gives the variance of the energy — and, converting from \(\beta\) to \(T\) via \(d\beta/dT=-1/(k_BT^2)\),
Two things follow immediately. First, since a variance can never be negative, \(C_V\ge0\) is now a proven statistical fact rather than an empirical assumption bolted onto thermodynamics. Second — and this is the real payoff — \(C_V\), which thermodynamics can only ever treat as an input parameter to be measured, is now calculable from first principles: give the energy levels, and the fluctuation formula hands back the specific heat directly.
Connecting back to thermodynamics: \(F = -k_BT\ln Z\)#
Apply the same peaked-product argument used for \(\Omega(E)\Omega'(E')\) in Lecture 20–21 directly to the integral defining \(Z\): \(\rho(\varepsilon)\) increases rapidly, \(e^{-\beta\varepsilon}\) decreases rapidly, so the integrand is sharply peaked near some \(\bar\varepsilon\), and to excellent approximation
using Boltzmann’s formula \(S=k_B\ln\Omega\). But \(U-TS\) is exactly the Helmholtz free energy \(F\) from Lecture 22–23 — so
This single formula is the bridge between microscopic dynamics (\(Z\), computed from the system’s own energy levels) and macroscopic thermodynamics (every quantity derivable from \(F\) by differentiation) — and testing it against a known result is the cleanest way to confirm everything above is self-consistent.
Worked example: the classical ideal gas, derived from \(Z\)#
Take \(N\) independent, point particles of mass \(m\) in volume \(V\), in contact with a bath at temperature \(T\). Since interactions are neglected, the total energy is additive, \(E = \sum_i \varepsilon_i\), and the exponential Boltzmann weight turns this sum into a product: \(Z = z_1^N\), where \(z_1\) is the single-particle partition function,
a standard Gaussian integral. Then \(F = -Nk_BT\big[\ln V + \tfrac32\ln(2\pi mk_BT/h^2)\big]\), and the pressure follows by direct differentiation,
the classical ideal-gas law, recovered from first principles — exactly the check that validates \(F=-k_BT\ln Z\) and everything built on it. The average energy follows just as directly, \(U=-\partial\ln Z/\partial\beta = \tfrac32 Nk_BT\): each of the three momentum components contributes \(\tfrac12k_BT\), the equipartition theorem for any degree of freedom that enters the Hamiltonian quadratically — the same \(\tfrac12\) traces back to the Gaussian integral above, and would become \(\tfrac14k_BT\) for a quartic term instead, as in an anharmonic oscillator’s potential energy. Combining \(U\) and \(P\) gives \(P = \tfrac23\,U/V\) — the \(2\) from the quadratic dispersion, the \(3\) from the spatial dimension, a relation that survives (with different numbers) even for quantum gases; for photons, with linear dispersion, it becomes \(P=U/3V\).
The Gibbs paradox and its fix#
Differentiating \(F\) for \(S=-\partial F/\partial T\) produces a term \(Nk_B\ln V\) — but physically, doubling \(V\), \(N\), and nothing else at fixed \(T\) should exactly double \(S\); a bare, un-divided \(V\) inside the logarithm wrecks that extensivity. The culprit is overcounting: treating the \(N\) particles as distinguishable counts every permutation of identical particles among occupied states as a different microstate, when physically it is the same one. The fix — valid whenever the number of available single-particle states vastly exceeds \(N\), so that two particles landing in the same state is negligibly rare — is to divide by \(N!\):
Using Stirling’s approximation, \(N!\approx N^Ne^{-N}\), the correction effectively replaces \(V \to V/N\) inside the logarithm — restoring exactly the extensive dependence that was missing, and giving the properly extensive Sackur–Tetrode entropy. Two compensating “mistakes” — dividing by \(N!\) and still summing every particle over every state independently, rather than tracking occupation numbers exactly — cancel to excellent approximation whenever states vastly outnumber particles, which is the regime this whole calculation assumed from the start.
When classical statistics is (and isn’t) valid#
That last caveat can be made quantitative. Quantum mechanically, a particle is not a point but a wave packet of width set by its thermal de Broglie wavelength, \(\lambda \sim h/p_{\rm rms}\), with \(p_{\rm rms}\sim\sqrt{mk_BT}\) from equipartition. Treating particles as distinguishable, classical points is only sensible if the mean interparticle spacing \(r_s=(V/N)^{1/3}\) is much larger than this spread — otherwise wave packets overlap and “which particle is which” stops being a meaningful question:
the degeneracy parameter. For nitrogen at room temperature and pressure this comes out around \(10^{-7}\) — deep in the classical regime, which is exactly why the ideal-gas derivation above works so well for ordinary air. The same estimate flips the other way for electrons in a metal, for matter in a neutron star, and for helium approaching absolute zero — systems where \(n\lambda^3 \sim 1\) or larger, and classical (Maxwell–Boltzmann) statistics breaks down entirely.
The single-particle energy distribution \(P(\varepsilon)\propto\rho(\varepsilon)\, e^{-\beta\varepsilon}\) is the product of the rising density of states \(\rho(\varepsilon) \propto\sqrt\varepsilon\) from Lecture 22–23 and the falling Boltzmann factor — the same peaked-product mechanism used twice already, now visualized directly. As \(T\) rises, the peak (at \(\varepsilon=k_BT/2\)) shifts outward and the whole distribution broadens, exactly the classical Maxwell energy distribution underlying the Maxwell–Boltzmann speed distribution of an ordinary gas.
This is also where Balakrishnan’s own lecture pauses: the natural next step is quantum statistics — replacing this particle-by-particle (column-wise) tally with an occupation-number (row-wise) count of how many particles sit in each state, the change that produces Bose–Einstein and Fermi–Dirac statistics in place of Maxwell–Boltzmann, and becomes unavoidable exactly where the degeneracy parameter \(n\lambda^3\) stops being small.