The Canonical Ensemble and the Boltzmann Distribution

The microcanonical ensemble fixes the energy exactly and counts the microstates consistent with it. That description is natural for an isolated system but awkward for the systems actually met in a laboratory, which sit in thermal contact with their surroundings and hold a fixed temperature rather than a fixed energy. The canonical ensemble is the description adapted to that situation: the system of interest exchanges energy freely with a large reservoir, its energy fluctuates, and what is held constant is the temperature the reservoir imposes. The central result is that the probability of finding the system in a definite microstate depends on that microstate only through its energy, and does so through a single exponential, the Boltzmann factor.

System and reservoir

Let a system $S$ be in thermal contact with a reservoir $R$ much larger than itself, the two enclosed by rigid adiabatic walls so that the combined system $S \cup R$ is isolated with fixed total energy $E_{tot}$ . Energy passes freely across the internal diathermal wall, so the system energy is not fixed; only the sum $E_{S} + E_{R} = E_{tot}$ is.¹ The reservoir is defined by being so large that any energy the system draws from it changes its temperature negligibly: $C_{R} \to \infty$ , and $E_{tot} \approx E_{R}$ .

The combined system is isolated, so the microcanonical postulate applies to it: every microstate of $S \cup R$ consistent with the total energy is equally likely. Fix attention on one definite microstate $i$ of the system, with energy $E_{i}$ . The reservoir must then hold energy $E_{tot} - E_{i}$ , and the number of combined microstates in which the system occupies exactly state $i$ equals the number of reservoir microstates at that energy, $Ω_{R} (E_{tot} - E_{i})$ . By equal a priori probability, the probability of the system microstate $i$ is proportional to that reservoir count,

p_{i} \propto Ω_{R} (E_{tot} - E_{i}) .

The system multiplicity does not appear: $i$ is a single specified microstate, weighted only by how many ways the reservoir can supply the complementary energy.

A small system exchanges energy with a large reservoir across a diathermal wall; the total energy is fixed, and each system microstate of energy

E_{i}

is weighted by the reservoir's multiplicity at the complementary energy.

Expanding the reservoir entropy

The reservoir count is enormous and varies rapidly with energy, so it is the logarithm that is well-behaved. Write it through the reservoir entropy, $k_{B} ln Ω_{R} (E_{R}) = S_{R} (E_{R})$ , and expand $S_{R}$ about the full energy $E_{tot}$ in powers of the small system energy $E_{i}$ :

ln Ω_{R} (E_{tot} - E_{i}) = \frac{1}{k _{B}} S_{R} (E_{tot} - E_{i}) = \frac{1}{k _{B}} [S_{R} (E_{tot}) - E_{i} (\frac{\partial S _{R}}{\partial E _{R}}) + \frac{1}{2} E_{i}^{2} (\frac{\partial ^{2} S _{R}}{\partial E _{R}^{2}}) - \dots] .

The first derivative is the reservoir's inverse temperature, by the thermodynamic definition established in the microcanonical ensemble,

(\frac{\partial S _{R}}{\partial E _{R}})_{E_{tot}} = \frac{1}{T} .

The second derivative measures how the reservoir temperature responds to energy loss: $\partial^{2} S_{R} / \partial E_{R}^{2} = - 1/ (T^{2} C_{R})$ , where $C_{R}$ is the reservoir heat capacity. For a reservoir this term and all higher ones scale as inverse powers of $C_{R}$ and vanish in the limit $C_{R} \to \infty$ .² The expansion collapses to its linear term,

ln Ω_{R} (E_{tot} - E_{i}) = \frac{S _{R} ( E _{tot} )}{k _{B}} - \frac{E _{i}}{k _{B} T} + O (\frac{1}{C _{R}}) .

Exponentiating, the constant $S_{R} (E_{tot})$ produces an $E_{i}$ -independent prefactor absorbed into normalization, and the probability of system microstate $i$ is

p_{i} = \frac{1}{Z} e^{- E_{i} / k_{B} T} = \frac{1}{Z} e^{- β E_{i}}, β \equiv \frac{1}{k _{B} T} .

The distribution is a statement about microstates, not energy levels. If several microstates share an energy $E$ — a degeneracy $g (E)$ — the probability that the system has that energy is $g (E) e^{- β E} / Z$ , the Boltzmann factor weighted by the number of states carrying it. This distinction between a single microstate and an energy level with its degeneracy is the source of every entropy term that follows.

The linearization is the entire physical content. Only the first derivative of the reservoir entropy survives, and that derivative is the temperature; the reservoir influences the system through one number. The ratio of probabilities of two microstates,

\frac{p _{i}}{p _{j}} = e^{- β (E_{i} - E_{j})},

depends only on their energy difference and the temperature, never on the reservoir's internal structure or on $E_{tot}$ .

The reservoir entropy

S_{R} (E_{tot} - E_{i})

is nearly linear in the small system energy

E_{i}

; the slope

- 1/ T

sets the Boltzmann factor, and the curvature scales as

1/ C_{R}

and vanishes for a large reservoir.

Maximum entropy at fixed mean energy

A second derivation reaches the same distribution without invoking a reservoir at all, by asking which probability assignment over microstates is least committal given a known mean energy. This is the maximum-entropy route, and it exposes $β$ as the multiplier enforcing the energy constraint.³

Take the Gibbs entropy of an arbitrary distribution ${p_{i}}$ over the system microstates,

S = - k_{B} i \sum p_{i} ln p_{i},

and maximize it subject to two constraints: normalization $\sum_{i} p_{i} = 1$ and a fixed mean energy $\sum_{i} p_{i} E_{i} = ⟨ E ⟩$ . Introduce Lagrange multipliers $α$ and $β$ for the two constraints and extremize the functional

L = - k_{B} i \sum p_{i} ln p_{i} - k_{B} α (i \sum p_{i} - 1) - k_{B} β (i \sum p_{i} E_{i} - ⟨ E ⟩) .

Setting $\partial L / \partial p_{i} = 0$ gives, for each microstate,

- ln p_{i} - 1 - α - β E_{i} = 0 ⟹ p_{i} = e^{- 1 - α} e^{- β E_{i}} .

Normalization fixes the prefactor, $e^{- 1 - α} = 1/ Z$ with $Z = \sum_{i} e^{- β E_{i}}$ , so $p_{i} = e^{- β E_{i}} / Z$ : the Boltzmann distribution again. The extremum is a maximum because $- \sum_{i} p_{i} ln p_{i}$ is concave in ${p_{i}}$ , so the stationary point is unique and is the global maximum.

The second multiplier $β$ is fixed by matching to thermodynamics. The maximum value of the Gibbs entropy, evaluated on the Boltzmann distribution, obeys $d S = k_{B} β d ⟨ E ⟩$ at fixed constraints, and comparison with the thermodynamic relation $d S = d U / T$ at constant volume gives $β = 1/ k_{B} T$ .⁴ The two derivations agree: the reservoir imposes the same distribution that maximum-entropy inference selects, and in both $β$ is the inverse temperature in energy units.

The Gibbs entropy of a two-microstate system,

- k_{B} [p ln p + (1 - p) ln (1 - p)]

, is concave with a single maximum; a fixed mean energy shifts the extremum to the Boltzmann weights rather than the equal-probability peak.

The temperature scale of the exponential

The Boltzmann factor sets an energy scale $k_{B} T$ against which every level gap is measured. States with $E_{i} - E_{0} ≪ k_{B} T$ above the ground state are populated nearly as heavily as the ground state; states with $E_{i} - E_{0} ≫ k_{B} T$ are exponentially suppressed. Temperature is the width of the accessible band: raising $T$ flattens the exponential and spreads probability up the spectrum, lowering it concentrates the system into its lowest states.

The Boltzmann factor

e^{- β E}

against microstate energy at two temperatures; the higher temperature (shallower curve) spreads probability up the spectrum, the lower temperature concentrates it near the ground state.

The Boltzmann distribution built here is the working tool of the rest of the module. The normalizing sum $Z$ , treated so far as a bookkeeping constant, turns out to encode all of the thermodynamics: differentiating it produces the mean energy, entropy, pressure, and the free energy, which is the subject of the next lesson.

Summary

A system in thermal contact with a large reservoir at temperature $T$ is described by the canonical ensemble: the total $S \cup R$ is isolated, so each system microstate $i$ is weighted by the reservoir multiplicity $Ω_{R} (E_{tot} - E_{i})$ .
Expanding the reservoir entropy to first order in $E_{i}$ keeps only the slope $\partial S_{R} / \partial E_{R} = 1/ T$ ; higher terms scale as $1/ C_{R}$ and vanish for a large reservoir, giving $p_{i} = e^{- β E_{i}} / Z$ with $β = 1/ k_{B} T$ .
Maximizing the Gibbs entropy $- k_{B} \sum_{i} p_{i} ln p_{i}$ at fixed mean energy yields the same distribution, with $β$ the Lagrange multiplier for the energy constraint; the canonical distribution is the least biased one consistent with a known $⟨ E ⟩$ .
The Boltzmann factor measures every level gap against the energy scale $k_{B} T$ : states within $k_{B} T$ of the ground state are appreciably populated, states far above it are exponentially rare, and raising $T$ spreads probability up the spectrum.

Reif, Fundamentals of Statistical and Thermal Physics, §6.1–6.2 — the ensemble of a system in contact with a heat bath, and the derivation of the canonical distribution from the microcanonical postulate applied to the combined system. Full text via the Waveland reprint (2009). ↩
Reif, Fundamentals of Statistical and Thermal Physics, §6.2, and Kardar, Statistical Physics of Particles, §4.6 — the Taylor expansion of the reservoir entropy and the argument that curvature terms are suppressed by the reservoir heat capacity. Kardar's treatment: MIT OCW 8.333, https://ocw.mit.edu/courses/8-333-statistical-mechanics-i-statistical-mechanics-of-particles-fall-2013/. ↩
Kardar, Statistical Physics of Particles, §4.6, and Pathria & Beale, Statistical Mechanics (4th ed.), §3.2 — the canonical distribution as the maximum-entropy assignment at fixed mean energy, with $β$ the multiplier conjugate to the energy constraint. ↩
Schroeder, An Introduction to Thermal Physics, §6.1 — the Boltzmann factor from the multiplicity of the reservoir, and the identification $β = 1/ k_{B} T$ by matching to the thermodynamic definition of temperature. Companion material at https://physics.weber.edu/schroeder/thermal/. ↩