Derivation of the Canonical Distribution from the Microcanonical Ensemble

The microcanonical ensemble is the bedrock: an isolated system of fixed energy, with the single postulate that every accessible microstate is equally likely. The canonical ensemble is the workhorse: a system at fixed temperature, with microstate probabilities weighted by the Boltzmann factor $e^{-\beta E}$ . These look like two different theories — one democratic (all states equal), the other exponentially biased. This article shows they are the same theory: the Boltzmann distribution is derived, not postulated, from the microcanonical bedrock, simply by taking a small system and putting it in contact with a large one. The trick is a change of viewpoint. Take a small system $S$ and surround it with a huge reservoir $R$ . Together, $S+R$ form an isolated whole with fixed total energy — a microcanonical system, to which the equal-a-priori postulate applies. Now ask: what is the probability that $S$ sits in a specific microstate $i$ ? It is proportional to the number of ways the reservoir can accommodate the leftover energy $E_{\text{tot}}-E_i$ . Because the reservoir is enormous, its entropy is a smooth function of energy, and expanding it to first order in the small system energy $E_i$ produces the exponential: $\Omega_R(E_{\text{tot}}-E_i)\propto e^{-\beta E_i}$ , with $\beta$ the reservoir's inverse temperature. That is the entire content of the canonical ensemble, and it explains why temperature appears where energy was: the reservoir's $\beta=(\partial S_R/\partial E)/k_B=1/k_BT$ imprints itself on the system. The derivation also reveals its own limits: the linear (first-order) expansion is exact only because the second-order term is suppressed by the reservoir's size, $\sim N_S/N_R\to0$ . This article carries out the reservoir expansion carefully, identifies $\beta$ as inverse temperature, quantifies why the expansion is exact, and checks the whole thing numerically on two coupled Einstein solids where the small one is driven to a Boltzmann distribution by the large one.

Key takeaways

The canonical (Boltzmann) distribution is derived from the microcanonical postulate by coupling a small system $S$ to a large reservoir $R$ and treating $S+R$ as isolated.
The probability of system microstate $i$ is $\propto\Omega_R(E_{\text{tot}}-E_i)$ — the number of reservoir states compatible with it.
Expanding the reservoir entropy (not $\Omega_R$ ) to first order: $\ln\Omega_R(E_{\text{tot}}-E_i)\approx\text{const}-\beta E_i$ , so $p_i\propto e^{-\beta E_i}$ .
The coefficient is inverse temperature: $\beta=\left.\partial\ln\Omega_R/\partial E\right|_{E_{\text{tot}}}=1/k_BT$ — the reservoir imprints its temperature on the system.
The expansion is exact because the second-order term is suppressed by $\sim N_S/N_R\to0$ : only an infinite reservoir gives a pure exponential.
Key idea: the Boltzmann distribution is not an axiom but a consequence — expanding a large reservoir's entropy to first order in the system energy turns the microcanonical postulate into $p_i\propto e^{-\beta E_i}$ .

The core insight

Coupling a small system to a large reservoir and applying the microcanonical equal-a-priori postulate to the isolated whole yields the canonical distribution: $p_i\propto\Omega_R(E_{\text{tot}}-E_i)$ , and expanding the reservoir entropy to first order gives $p_i\propto e^{-\beta E_i}$ with $\beta=1/k_BT$ . The exponential is exact because the second-order term vanishes as $N_S/N_R\to0$ .

Where this lives today

This derivation is the logical spine that connects the two most-used ensembles, and its ideas recur wherever "temperature control" must be implemented. In molecular-dynamics simulations, a real system of a few thousand atoms is coupled to a thermostat — a mathematical reservoir (Nosé–Hoover, Langevin, Andersen) — precisely to sample the canonical distribution $e^{-\beta E}$ rather than the microcanonical one; the thermostat algorithms are engineered reservoirs, and understanding their subtle finite-size artifacts requires exactly the second-order corrections this article quantifies. Getting the reservoir coupling wrong biases every computed free energy. The reservoir picture also underlies the modern theory of open quantum systems and quantum thermodynamics. A qubit in contact with a thermal bath relaxes to the quantum canonical state $\rho=e^{-\beta\hat H}/Z$ by exactly the reasoning here, generalized to density matrices; the theory of thermalization (the eigenstate thermalization hypothesis) asks how a large isolated quantum system can act as its own reservoir for its subsystems, deriving canonical behaviour from microcanonical (unitary) dynamics. This is one of the deepest current questions in statistical physics: how the Boltzmann distribution — here derived by coupling to an external bath — emerges for a subsystem of a closed, isolated quantum world. The humble reservoir expansion is the classical shadow of that profound quantum-thermalization program, and it remains the cleanest demonstration that the exponential Boltzmann weight is not a postulate but a consequence of counting states.

The physical picture

Start from the one postulate of the microcanonical ensemble: for an isolated system of fixed energy, all accessible microstates are equally likely. Now the small system we care about is not isolated — it exchanges energy with its surroundings. So enlarge the box: consider system + reservoir together as one isolated whole. That combined system is microcanonical, and the postulate applies to it. Every microstate of the whole — every way of splitting the fixed total energy between $S$ and $R$ — is equally likely.

Figure

System S in contact with a large reservoir R at total energy E_tot. When S is in a definite state of energy E, the reservoir holds E_tot − E and can be in Ω_R(E_tot − E) states.

Here is the pivot. Fix the small system in a definite microstate $i$ of energy $E_i$ — that is one state of $S$ , so it carries no multiplicity of its own. But the reservoir can then be in any of $\Omega_R(E_{\text{tot}}-E_i)$ microstates, all equally likely. So the probability that $S$ is in state $i$ is proportional to $\Omega_R(E_{\text{tot}}-E_i)$ : the system prefers low-energy states not because they are intrinsically favoured, but because leaving more energy for the reservoir gives the reservoir more ways to arrange itself. The exponential bias of the canonical ensemble is really the reservoir's entropy talking. Why an exponential, specifically? Because $\Omega_R$ is astronomically steep, but its logarithm — the entropy — is smooth and nearly linear over the tiny energy range the small system samples. Expand $\ln\Omega_R$ to first order and you get $\text{const}-\beta E_i$ , so $\Omega_R\propto e^{-\beta E_i}$ . The punchline plot below shows this: the reservoir's entropy curve is gently sloped, and its local slope $\beta=\partial\ln\Omega_R/\partial E$ — evaluated at $E_{\text{tot}}$ — is exactly the inverse temperature that the reservoir imprints on the system. Temperature is the slope of the reservoir's entropy.

Figure

The reservoir entropy ln Ω_R(E_tot − E) is smooth; its slope at E_tot is β = 1/k_BT. Linearizing gives ln Ω_R ≈ const − βE, hence p ∝ e^{-βE}.

The core equations

The starting point is the microcanonical probability of a system microstate, proportional to the reservoir multiplicity:

p_i \propto \Omega_R(E_{\text{tot}}-E_i) = \exp\!\left[\frac{1}{k_B}S_R(E_{\text{tot}}-E_i)\right] \qquad (1)

where: $p_i$ — probability of system microstate $i$ ; $\Omega_R$ — reservoir multiplicity; $S_R=k_B\ln\Omega_R$ — reservoir entropy; $E_{\text{tot}}=E_S+E_R$ — fixed total energy. The reservoir entropy in the exponent is what will be expanded. Expanding the reservoir entropy (smooth, unlike $\Omega_R$ ) to first order in the small energy $E_i$ :

S_R(E_{\text{tot}}-E_i) = S_R(E_{\text{tot}}) - E_i\left.\frac{\partial S_R}{\partial E}\right|_{E_{\text{tot}}} + \tfrac12 E_i^2\left.\frac{\partial^2 S_R}{\partial E^2}\right|+\cdots \qquad (2)

where: the first-order term is linear in $E_i$ ; the second-order term is the curvature. The reservoir's largeness is what makes the linear term dominate. Identifying the linear coefficient as inverse temperature gives the Boltzmann distribution:

\frac{1}{k_B}\frac{\partial S_R}{\partial E}=\frac{1}{k_BT}=\beta \quad\Rightarrow\quad p_i\propto e^{-\beta E_i},\qquad p_i=\frac{e^{-\beta E_i}}{Z} \qquad (3)

where: $\beta=1/k_BT$ — the reservoir's inverse temperature (from $1/T=\partial S/\partial E$ ); $Z=\sum_i e^{-\beta E_i}$ — the normalizing partition function. The canonical distribution is derived, not assumed. The second-order term measures the error, and it is suppressed by the reservoir size:

\frac{\text{2nd-order}}{\text{1st-order}} \sim \frac{E_i}{E_R} \sim \frac{N_S}{N_R} \xrightarrow{N_R\to\infty} 0 \qquad (4)

where: $N_S,N_R$ — number of degrees of freedom in system and reservoir. The curvature term (related to the reservoir's finite heat capacity) vanishes as the reservoir grows, so the exponential is exact only for an infinite reservoir.

Derivation ADeriving the Boltzmann distribution and identifying β as inverse temperature

The goal is to derive the canonical probability $p_i\propto e^{-\beta E_i}$ from the microcanonical postulate applied to system-plus-reservoir, and to show that $\beta$ is exactly the reservoir's inverse temperature. The strategy is to expand the reservoir entropy (not the steep multiplicity) to first order and identify the linear coefficient thermodynamically. We use the ideal-gas reservoir to make the coefficient explicit.

Step 1. By the equal-a-priori postulate for the isolated whole, $p_i\propto\Omega_R(E_{\text{tot}}-E_i)$ . Write this as $\exp[S_R(E_{\text{tot}}-E_i)/k_B]$ so we can expand the smooth entropy rather than the astronomically steep $\Omega_R$ . Why: $\Omega_R$ varies by factors of $e^{10^{23}}$ over tiny energy ranges, so a Taylor expansion of $\Omega_R$ diverges; the entropy $S_R=k_B\ln\Omega_R$ is smooth and extensive, and its expansion converges.

Step 2. Taylor-expand: $S_R(E_{\text{tot}}-E_i)=S_R(E_{\text{tot}})-E_i(\partial S_R/\partial E)+\cdots$ . The constant $S_R(E_{\text{tot}})$ is absorbed into the normalization, leaving $p_i\propto\exp[-E_i(\partial S_R/\partial E)/k_B]$ . Why: only the $E_i$ -dependent term matters for the relative probabilities; the constant reservoir entropy at the full energy is the same for every system state.

Step 3. Identify the coefficient. The thermodynamic definition of temperature is $1/T=(\partial S/\partial E)_{V,N}$ , so $(\partial S_R/\partial E)/k_B=1/(k_BT)=\beta$ . Hence $p_i\propto e^{-\beta E_i}$ , with $\beta$ the reservoir's inverse temperature. For a monatomic ideal-gas reservoir, $S_R=\tfrac32 N_R k_B\ln E_R$ gives $\partial S_R/\partial E=\tfrac32 N_R k_B/E_R$ , and with $E_R=\tfrac32 N_R k_BT$ this is exactly $1/T$ — a concrete check. Why: the slope of the reservoir's entropy versus energy is the inverse temperature; the reservoir stamps its $\beta$ onto the small system's distribution.

Step 4. Normalize: $p_i=e^{-\beta E_i}/Z$ with $Z=\sum_i e^{-\beta E_i}$ . This is the canonical ensemble, obtained purely from the microcanonical postulate plus the largeness of the reservoir. Why: dividing by the sum enforces $\sum_i p_i=1$ ; the whole canonical formalism is the microcanonical postulate seen from the small system's perspective. *Verified: for the ideal-gas reservoir, $\partial\ln\Omega_R/\partial E=\tfrac32 N_R/E_{\text{tot}}=1/(k_BT)$ with $E_{\text{tot}}=\tfrac32 N_R k_BT$ , confirmed symbolically in verify_aq-sm-canonical-from-microcanonical.py.*

Go deeper · Derivation B · 2 minWhy the linear expansion is exact — the owned result on the second-order term

Derivation BWhy the linear expansion is exact — the owned result on the second-order term

This is the article's owned structural result: the reservoir expansion is not merely a convenient approximation but becomes exact in the reservoir limit, because the second-order (curvature) term is suppressed by the ratio $N_S/N_R$ and vanishes as the reservoir grows. The goal is to compute the second-order term explicitly and show it dies. The strategy is to keep the quadratic term in the entropy expansion and estimate its size relative to the linear term for an ideal-gas reservoir.

Step 1. Keep two terms: $\ln\Omega_R(E_{\text{tot}}-E)=\ln\Omega_R(E_{\text{tot}})-\beta E+\tfrac12 c_2 E^2+\cdots$ , where $c_2=\partial^2\ln\Omega_R/\partial E^2$ . For the ideal-gas reservoir $\ln\Omega_R\propto\tfrac32 N_R\ln E$ , so $\beta=\tfrac32 N_R/E_{\text{tot}}$ and $c_2=-\tfrac32 N_R/(2E_{\text{tot}}^2)$ (negative — the entropy is concave). Why: differentiating the reservoir entropy twice gives a curvature inversely proportional to $E_{\text{tot}}^2$ and proportional to $N_R$ — the reservoir's finite heat capacity.

Step 2. Form the ratio of the quadratic to the linear term at a typical system energy $E\sim E_S$ : $\dfrac{|c_2|E^2/2}{\beta E}=\dfrac{|c_2|E}{2\beta}=\dfrac{E}{2E_{\text{tot}}}$ . With $E_S\sim\tfrac32 N_S k_BT$ and $E_{\text{tot}}\sim\tfrac32 N_R k_BT$ , this is $\dfrac{N_S}{2N_R}$ . Why: the two energy scales are set by the numbers of degrees of freedom, so their ratio is the ratio of system to reservoir sizes — small when the reservoir is large.

Step 3. Take the reservoir limit $N_R\to\infty$ at fixed $N_S$ and $T$ : the ratio $N_S/2N_R\to0$ , so the quadratic (and all higher) terms vanish and $\ln\Omega_R(E_{\text{tot}}-E)\to\text{const}-\beta E$ exactly. The Boltzmann distribution is exact for an infinite reservoir, not merely approximate. Why: an infinite reservoir has infinite heat capacity, so absorbing the system's energy does not change its temperature at all — the slope $\beta$ is truly constant, and the expansion terminates at first order.

Step 4. Physical reading of the corrections. For a finite reservoir the residual curvature reflects the reservoir cooling slightly as it absorbs energy; this is the origin of small canonical-vs-microcanonical differences that scale as $1/N_R$ . Why: a finite bath has a finite heat capacity, so its temperature is not perfectly rigid; the second-order term encodes exactly this back-reaction, which disappears only in the idealized infinite-reservoir limit. *Verified: for the ideal-gas reservoir the 2nd/1st-order term ratio equals $N_S/(2N_R)\to0$ ; two coupled Einstein solids (small $N_S=4$ , large $N_R=4000$ ) drive the small system to a Boltzmann distribution — the slope of $\ln(P/\Omega_S)$ matches $-\beta$ and $\langle q_S\rangle$ matches the canonical $N_S/(e^\beta-1)$ within $10\%$ , in the verify script.*

Limiting cases

The derivation across regimes:

Case	Result	Meaning
Infinite reservoir	$p_i = e^{-\beta E_i}/Z$ exactly	The pure canonical ensemble; reservoir temperature perfectly rigid.
Finite reservoir	corrections $\sim N_S/N_R$	Small deviations from Boltzmann; the bath cools slightly as it gives up energy.
System = reservoir size	expansion invalid	No separation of scales; must treat the whole microcanonically.
Ideal-gas reservoir	$\beta = \tfrac32 N_R/E_{\text{tot}}$	The slope of $S_R$ vs $E$ is explicitly $1/k_BT$ .
Zero coupling	isolated system	Microcanonical; no energy exchange, no Boltzmann weighting.

Common trap“You expand the reservoir multiplicity Ω_R to first order in E”

No — you must expand the entropy $\ln\Omega_R$ , never $\Omega_R$ itself. The multiplicity $\Omega_R$ is an absurdly steep function — it varies by factors of $e^{10^{23}}$ over the tiny energy range the small system explores — so a Taylor expansion of $\Omega_R$ around $E_{\text{tot}}$ is hopeless: the "small" correction dwarfs the leading term. The entropy $S_R=k_B\ln\Omega_R$ , by contrast, is smooth, extensive, and nearly linear over that range, so its expansion converges beautifully, and $\ln\Omega_R\approx\text{const}-\beta E$ gives $\Omega_R\approx\text{const}\cdot e^{-\beta E}$ . This is the single most important technical point in the derivation, and it is why we work with entropy (the log) rather than multiplicity. Two related subtleties: (1) $\beta$ is a property of the reservoir, evaluated at the full energy $E_{\text{tot}}$ , not of the small system; and (2) the whole scheme requires a separation of scales ( $N_S\ll N_R$ ) — if the "reservoir" is comparable to the system, there is no valid expansion and the canonical ensemble does not emerge.

Worked examples

1 · Two coupled Einstein solids drive the small one to Boltzmann (a real emergence)

A small Einstein solid ( $N_S=4$ oscillators) is coupled to a large one ( $N_R=4000$ ) sharing $q_{\text{tot}}=4000$ energy quanta. Show that the small system's energy distribution is Boltzmann, and find its effective temperature (Einstein solid $\Omega(N,q)=\binom{q+N-1}{q}$ ).

Step 1. The probability the small solid holds $q$ quanta is $P(q)\propto\Omega_S(q)\,\Omega_R(q_{\text{tot}}-q)$ , by the microcanonical postulate on the combined solid. Enumerate: $\Omega_S(q)=\binom{q+3}{q}$ , $\Omega_R(4000-q)=\binom{4000-q+3999}{4000-q}$ .

Step 2. The reservoir's inverse "temperature" per quantum is $\beta=\partial\ln\Omega_R/\partial q\approx\ln(1+N_R/q_R)$ . With $q_R\approx q_{\text{tot}}=4000$ and $N_R=4000$ : $\beta = \ln(1+4000/4000) = \ln 2 = \mathbf{0.693}$ per quantum.

Step 3. Predicted Boltzmann distribution: $P(q)\propto\Omega_S(q)\,e^{-\beta q}$ . Numerically (in the verify script) the slope of $\ln[P(q)/\Omega_S(q)]$ against $q$ is $-\beta\approx-0.69$ , confirming the small system samples its states with a Boltzmann weight set by the reservoir — even though nothing was postulated about temperature.

Step 4. Mean energy: canonically $\langle q_S\rangle=N_S/(e^\beta-1)=4/(e^{0.693}-1)=4/1.0=\mathbf{4.0}$ quanta, matching the direct microcanonical average within a few percent. A large "dumb" reservoir, obeying only equal-a-priori probabilities, forces a small system into the exponential Boltzmann distribution — the canonical ensemble emerging from the microcanonical, computed explicitly.

2 · Identifying β for an ideal-gas reservoir (exam level)

A small system is in contact with a monatomic ideal-gas reservoir of $N_R$ atoms and total energy $E_{\text{tot}}$ (with $E_S\ll E_{\text{tot}}$ ). (a) Compute $\beta=\partial\ln\Omega_R/\partial E$ . (b) Verify it equals $1/k_BT$ . (c) Estimate the size of the neglected second-order term for $N_S=1$ atom, $N_R=10^{20}$ .

Step 1. Ideal-gas multiplicity scales as $\Omega_R\propto E^{3N_R/2}$ , so $\ln\Omega_R=\tfrac32 N_R\ln E+\text{const}$ and $\beta=\partial\ln\Omega_R/\partial E=\dfrac{3N_R}{2E_{\text{tot}}}$ .

Step 2. The reservoir energy is $E_{\text{tot}}=\tfrac32 N_R k_BT$ (equipartition). Substitute: $\beta=\dfrac{3N_R/2}{\tfrac32 N_R k_BT}=\dfrac{1}{k_BT}$ . Confirmed: the slope of the reservoir entropy is exactly the inverse temperature.

Step 3. Second-order term ratio $\approx\dfrac{N_S}{2N_R}=\dfrac{1}{2\times10^{20}}=\mathbf{5\times10^{-21}}$ . The curvature correction is one part in $2\times10^{20}$ — utterly negligible.

Step 4. Interpretation. For a macroscopic reservoir the Boltzmann distribution is exact to twenty-one decimal places; the "approximation" of truncating at first order is, for all practical purposes, an identity. Only for a pathologically small "reservoir" (comparable in size to the system) would the corrections matter — and then the canonical ensemble simply does not apply.

Origins · 1 minheat reservoir and how this was discovered

heat reservoir

statistical-mechanics concept · —

The heat reservoir (or heat bath) is the idealization at the heart of the canonical ensemble: a system so large that it can exchange arbitrary amounts of energy with a smaller system while its own temperature remains fixed. It is the thermodynamic analogue of an infinite ocean into which you can pour any amount of heat without warming it. Formally, a reservoir has infinite heat capacity, so its inverse temperature $\beta=\partial\ln\Omega_R/\partial E$ is constant — and it is precisely this constancy that makes the reservoir-entropy expansion terminate at first order, producing the pure exponential Boltzmann weight. The concept crystallized with Gibbs's ensemble formalism (1902): the canonical ensemble is the statistics of a system in contact with a heat reservoir. Every real thermostat — a water bath, a cryostat, the atmosphere, a computer's thermostat algorithm in molecular dynamics — is an approximation to an ideal reservoir. The reservoir is where the arrow of "temperature control" enters statistical mechanics, and understanding that the Boltzmann factor is the reservoir's entropy in disguise is the conceptual key to the entire canonical formalism.

History

1877

— Boltzmann's combinatorial entropy S = k log W provides the reservoir-counting foundation.

1902

— Gibbs's Elementary Principles derives the canonical ensemble from the reservoir picture and names it.

1912

— the Einstein/Debye solids provide the first explicit reservoir-coupling computations of specific heat.

1929

— Szilard analyzes the reservoir's role in the thermodynamics of information (Maxwell's demon).

1957

— Jaynes recasts the derivation as maximum entropy subject to a mean-energy constraint, an alternative route to the same e^{−βE}.

Going further

References. Reif, Fundamentals of Statistical and Thermal Physics, Ch. 6 (the classic reservoir derivation) · Pathria & Beale, Statistical Mechanics, 3rd ed., Ch. 3 · Kardar, Statistical Physics of Particles, §4.2 · Schroeder, An Introduction to Thermal Physics, Ch. 6 · Sethna, Statistical Mechanics, Ch. 6 · Landau & Lifshitz, Statistical Physics Part 1, §28.

Provenance. The reservoir derivation of the canonical distribution is standard (Reif Ch. 6; Schroeder Ch. 6). A verifier should reproduce: $p_i\propto\Omega_R(E_{\text{tot}}-E_i)$ ; the first-order entropy expansion $\ln\Omega_R\approx\text{const}-\beta E$ with $\beta=\partial\ln\Omega_R/\partial E=1/k_BT$ (checked on the ideal-gas reservoir $\beta=\tfrac32 N_R/E_{\text{tot}}$ with $E_{\text{tot}}=\tfrac32 N_R k_BT$ ); the second-order suppression $\sim N_S/2N_R\to0$ ; and the two-Einstein-solid emergence of the Boltzmann distribution ( $\langle q_S\rangle$ matches $N_S/(e^\beta-1)$ ). Numerical instances are recomputed in verify_aq-sm-canonical-from-microcanonical.py.

Builds on: The Microcanonical Ensemble and the Postulate of Equal a Priori Probabilities · Boltzmann Entropy $S=k_B\ln\Omega$ · Temperature, Pressure, and Chemical Potential as Entropy Derivatives. Leads to: Canonical Ensemble · Boltzmann Factor · Canonical Partition Function · Helmholtz Free Energy from the Partition Function · Energy Fluctuations in the Canonical Ensemble · Grand Canonical Ensemble.

Why does the small system prefer low-energy states if all microstates of the whole are equally likely?

This is the beautiful subtlety of the derivation. It is true that all microstates of the combined system-plus-reservoir are equally likely — no state is intrinsically favoured. But when you ask about the small system alone, you must count how many combined states correspond to each small-system state. If the small system takes a low energy, it leaves more energy for the reservoir, and a reservoir with more energy has vastly more microstates available (its multiplicity $\Omega_R$ rises steeply with energy). So low-energy states of the small system are compatible with more equally-likely reservoir arrangements, and are therefore more probable — not because they are preferred, but because they leave the reservoir more room. The exponential bias of the Boltzmann factor is entirely the reservoir's entropy talking: the small system "wants" to give energy to the reservoir because that maximizes the total number of arrangements, which is just entropy maximization in action.

Where exactly does temperature come from in this derivation?

Temperature enters as the slope of the reservoir's entropy versus energy, and nowhere else. When you expand the reservoir entropy $S_R(E_{\text{tot}}-E)$ to first order in the small system's energy $E$ , the linear coefficient is $-(\partial S_R/\partial E)$ . The thermodynamic definition of temperature is precisely $1/T=(\partial S/\partial E)$ , so that coefficient is $1/T$ (in units where you divide by $k_B$ , it is $\beta=1/k_BT$ ). This is profound: the small system has no temperature of its own in the derivation — temperature is a property of the reservoir, and it gets imprinted on the small system's distribution through the coupling. Two systems in contact with the same reservoir end up with the same $\beta$ regardless of their internal structure, which is exactly the zeroth law (thermal equilibrium is transitive) emerging from the reservoir picture. So temperature is not something the small system possesses; it is something the reservoir stamps onto it.

What is the difference between this derivation and just postulating the Boltzmann distribution?

Logically, everything. If you postulate $p_i=e^{-\beta E_i}/Z$ , you have to accept the exponential form and the meaning of $\beta$ on faith, and you have two independent postulates (microcanonical equal-a-priori and canonical Boltzmann) that might seem unrelated. This derivation shows they are the same postulate viewed from different vantage points: the canonical Boltzmann distribution is a theorem, provable from the single microcanonical postulate plus the mathematical fact that a large reservoir's entropy is smooth and nearly linear over the range the small system explores. It also explains the two things a postulate leaves mysterious: why the weight is exponential (because the reservoir entropy is linear, and $e^{\text{linear}}$ is exponential) and why $\beta$ is inverse temperature (because it is the slope of the reservoir entropy, which is $1/T$ by definition). A derivation that reduces two postulates to one and explains the form of the result is far more satisfying — and far more illuminating about the why — than a postulate.

Does this mean the canonical ensemble is only an approximation to the microcanonical one?

For a finite reservoir, yes — there are small corrections of order $N_S/N_R$ . But in the idealized reservoir limit ( $N_R\to\infty$ ), the canonical distribution is exact, and for any realistic macroscopic reservoir the corrections are so tiny (one part in $10^{20}$ or smaller) that "approximation" is a technically-true-but-misleading word. More importantly, the two ensembles give identical thermodynamic predictions for large systems — this is ensemble equivalence. The canonical energy fluctuates while the microcanonical is fixed, but the fluctuations are relatively negligible ( $\sim N^{-1/2}$ ), so every thermodynamic quantity (energy, entropy, free energy, pressure) comes out the same. So it is not that one ensemble is "right" and the other "approximate"; they are two equally valid descriptions that agree in the thermodynamic limit, connected by exactly this reservoir derivation. You choose whichever is more convenient — usually canonical, because fixed temperature is easier to work with than fixed energy.

Can a system be its own reservoir? What if there's nothing external?

This is one of the deepest questions in modern statistical mechanics, and the answer is a qualified yes. For a large isolated system, any small subsystem can be treated canonically, with the rest of the system acting as its reservoir — you simply designate part of the whole as "system" and the remainder as "bath," and the derivation goes through. This is how the Boltzmann distribution emerges for local degrees of freedom even in a completely closed, isolated system: the vast number of other degrees of freedom serve as the reservoir. In quantum mechanics this idea is formalized by the eigenstate thermalization hypothesis (ETH), which proposes that a single energy eigenstate of a large chaotic quantum system already looks thermal (canonical) when you examine any small subsystem, so an isolated quantum system can thermalize its own parts through unitary evolution alone — no external bath needed. This resolves an old puzzle: how can a closed system, whose total dynamics is reversible and never increases entropy, exhibit thermal (irreversible-looking) behaviour in its subsystems? The answer is that the rest of the system is the reservoir. So "there is nothing external" does not prevent canonical behaviour — it just means the reservoir is internal.

Why must the reservoir be much larger than the system?

Because the derivation relies on the reservoir's temperature being essentially unchanged when it exchanges energy with the system — and that requires the reservoir to have a much larger heat capacity, i.e. many more degrees of freedom. Quantitatively, the neglected second-order term in the entropy expansion scales as $N_S/N_R$ , the ratio of system to reservoir sizes. If the reservoir is much larger ( $N_R\gg N_S$ ), this ratio is tiny and the linear expansion — hence the pure exponential Boltzmann weight — is essentially exact. If the "reservoir" is comparable in size to the system, the ratio is order one, the second-order (curvature) term is significant, and the distribution is not a clean Boltzmann exponential: the effective temperature shifts as energy flows, because the small "reservoir" noticeably cools when it gives up energy. In that case there is no separation of scales, no fixed temperature, and you must treat the combined system microcanonically. So the largeness of the reservoir is exactly what guarantees a rigid temperature and a pure canonical distribution; it is the mathematical condition for "temperature control" to make sense.