The Role of Ensemble Choice in Statistical Mechanics — Epoche B2
When Ensembles Disagree: The Limits of Ensemble Equivalence A first course in statistical mechanics distils equilibrium into a slogan [1] : a system left to itself settles into the macrostate of maximum entropy. A companion slogan usually rides along with it — that the three standard ensembles (microcanonical, canonical, grand canonical) are merely different bookkeeping schemes for the same physics, interchangeable once the system is large. For ordinary matter both slogans are sound. But they quietly assume something that a great many interesting systems violate: that energy is additive . When it is not — most spectacularly for anything bound by gravity — the ensembles genuinely disagree, and a quantity that is flatly forbidden in one becomes routine in the other. This note builds the argument from the definitions up: what each ensemble actually fixes, why they normally coincide, the additivity assumption hiding beneath that coincidence, and the headline consequence of its failure — a negative heat capacity, which (contrary to a common misstatement) lives in the microcanonical description and is impossible in the canonical one. Two ways to fix a system: the microcanonical and canonical ensembles The microcanonical ensemble describes an isolated system with its energy $E$, volume $V$ and particle number $N$ all held fixed. Its master quantity is the number of microstates $\Omega(E,V,N)$ compatible with those constraints; Boltzmann's relation turns that count into an entropy, and the thermodynamic temperature is read off as the rate at which entropy responds to energy, $$ S(E,V,N)=k_B\ln\Omega(E,V,N),\qquad \frac{1}{T}=\left(\frac{\partial S}{\partial E}\right)_{V,N}, $$ where $k_B$ is Boltzmann's constant. Here $E$ is the independent variable and $T$ is derived from it. The canonical ensemble swaps their roles: the system is placed in thermal contact with a large reservoir at fixed temperature $T$, so energy is free to flow in and out and only its average is pinned. Summing the Boltzmann weight $e^{-\beta E_i}$ over every microstate $i$ defines the partition function $Z$, from which the Helmholtz free energy follows, $$ Z=\sum_i e^{-\beta E_i}=\int dE\,\Omega(E)\,e^{-\beta E},\qquad F=-k_BT\ln Z,\qquad \beta=\frac{1}{k_BT}. $$ The second form of $Z$ simply groups the microstates by energy: $\Omega(E)\,dE$ of them lie in the shell $[E,E+dE]$, each carrying weight $e^{-\beta E}$. The two ensembles thus start from the same density of states $\Omega(E)$ but ask different questions — one at fixed $E$, the other at fixed $T$. Why the two normally give the same answer Writing $\Omega=e^{S/k_B}$ exposes the competition buried inside $Z$: $$ Z=\int dE\,\exp\!\left[\frac{1}{k_B}\Big(S(E)-\frac{E}{T}\Big)\right]\approx \exp\!\left[-\frac{F(E^{*})}{k_BT}\right],\qquad \left.\frac{\partial S}{\partial E}\right|_{E^{*}}=\frac{1}{T}. $$ The exponent pits entropy, which grows with $E$, against the Boltzmann suppression $E/T$. For a macroscopic system the bracket is enormous, so the integral is overwhelmingly dominated by its sharpest peak — the energy $E^{*}$ that maximises the exponent (a saddle-point , or Laplace, approximation), with $F(E^{*})=E^{*}-T\,S(E^{*})$ the free energy evaluated there. Setting the derivative to zero returns exactly the microcanonical relation $\partial S/\partial E=1/T$: the most probable canonical energy $E^{*}$ is precisely the microcanonical energy belonging to that temperature. This identification is a Legendre transform, and it is why the ensembles agree — provided the peak really is a maximum. How sharp is it? The peak's width is the size of the energy fluctuations, and for a normal system $$ \frac{\Delta E}{\langle E\rangle}=\frac{\sqrt{k_BT^2\,C_V}}{\langle E\rangle}\sim\frac{1}{\sqrt{N}}\longrightarrow 0\quad(N\to\infty), $$ where $\langle E\rangle$ denotes the thermal average of the energy and $C_V=(\partial\langle E\rangle/\partial T)_V$ its heat capacity. Because $C_V$ and $\langle E\rangle$ both scale with $N$, the relative spread shrinks as $1/\sqrt{N}$: in the thermodynamic limit the canonical energy is pinned to a single value and the fixed-$E$ and fixed-$T$ pictures become indistinguishable. For a mole of gas, $N\sim10^{23}$ gives $\Delta E/\langle E\rangle\sim10^{-12}$ — utterly negligible. This convergence is the entire content of "ensemble equivalence", and it is where the standard treatments — Pathria and Beale's among them — stop. The load-bearing assumption: additivity Every step above rests on one premise. To speak of a system "in contact with a reservoir", or to let it trade energy with its surroundings, we must be able to split the total energy into a piece belonging to the system and a piece belonging to the rest, with a negligible coupling between them. This is additivity (or extensivity): cut a body in two and $E_{\text{total}}=E_1+E_2+E_{\text{interaction}}$, with $E_{\text{interaction}}$ ignorable. For short-range forces it is ignorable, because the interaction energy comes only from the thin layer straddling the cut: it scales with the surface, $\sim N^{2/3}$, and is dwarfed by the bulk, $\sim N$. The order of magnitude follows from summing a pair potential $\sim r^{-\alpha}$ over a $d$-dimensional body of radius $R$: $$ \varepsilon=\frac{U}{N}\sim\int^{R}\! r^{\,d-1-\alpha}\,dr\quad\begin{cases}\text{finite as }R\to\infty, & \alpha\gt d\quad(\text{short range}),\\[3pt] \text{grows with }R, & \alpha\le d\quad(\text{long range}).\end{cases} $$ Here $\varepsilon=U/N$ is the potential energy per particle and the factor $r^{\,d-1}$ is the volume of a spherical shell of radius $r$. When $\alpha\gt d$ the integral converges and $\varepsilon$ stays finite as the system grows: energy is additive and the reservoir argument holds. When $\alpha\le d$ the integral diverges — the energy is dominated by distant pairs and is non-additive . Gravity is the extreme case, $\alpha=1$ in $d=3$: every particle feels every other, no subsystem is ever weakly coupled t