Energy Fluctuations and the Equivalence of Ensembles

Fixing the temperature rather than the energy comes at a price: the energy is no longer sharp. A canonical system exchanges energy with its reservoir, so its energy fluctuates about the mean $U=⟨E⟩$ from one instant to the next. The size of those fluctuations is not a free parameter — it is set by the same partition function that fixes the mean, through a second derivative of $\ln Z$. The result ties the spread of the energy to a measurable response, the heat capacity, and shows that for a macroscopic system the fluctuations are so small that fixing the temperature is indistinguishable from fixing the energy. This is the reason the canonical and microcanonical ensembles agree.

The variance from a second derivative

The mean energy is $U=-\partial \ln Z/\partial \beta$. Differentiating once more in $\beta$ reaches the variance. Start from $⟨E⟩=Z^{-1}{\sum }_i{E}_i{e}^{-\beta E_i}$ and differentiate with respect to $\beta$, remembering that $Z$ itself depends on $\beta$:

The right side is minus the variance of the energy. Equivalently, the variance is the second logarithmic derivative,

The first derivative of $\ln Z$ gives the mean, the second gives the variance; $\ln Z$ is the cumulant generating function of the energy, with $\beta$ as the generating variable.¹ Because the variance is a sum of squares it is non-negative, which forces $\partial ⟨E⟩/\partial \beta \le 0$: the mean energy always decreases as $\beta$ increases, that is, as temperature falls.

The fluctuation–response identity

Convert the $\beta$-derivative to a temperature derivative. With $\partial /\partial \beta =-k_B{T}^2\partial /\partial T$ and the heat capacity at constant volume $C_V=(\partial U/\partial T{)}_V$,

The identity is the first instance of a general pattern: an equilibrium fluctuation of a quantity equals the response of that quantity to its conjugate field. Here the energy fluctuation equals the response of the energy to temperature. The same structure recurs for particle-number fluctuations and the compressibility, and for magnetization fluctuations and the susceptibility, in later modules.² Two consequences follow immediately. First, since $⟨\Delta E^2⟩\ge 0$, the heat capacity is non-negative, $C_V\ge 0$: a stability condition derived here from statistics rather than assumed. Second, a diverging heat capacity signals diverging energy fluctuations, the hallmark of a continuous phase transition.

Relative fluctuations and the thermodynamic limit

The absolute variance grows with system size, but the physically relevant quantity is the fluctuation relative to the mean. For a system of $N$ particles both $U$ and $C_V$ are extensive, $U\sim Nu$ and $C_V\sim N{c}_v$ with $u$ and $c_v$ intensive. The relative spread of the energy is therefore

The relative fluctuation shrinks as $1/\sqrt{N}$.³ For a macroscopic sample with $N\sim 10^{23}$ it is of order $10^{-12}$: the energy of a mole of gas at fixed temperature is constant to twelve significant figures. The distinction between fixing the temperature and fixing the energy is undetectable at that scale, which is the quantitative statement that the ensembles agree.

The sharpness of the energy distribution

The relative-fluctuation estimate can be sharpened into the full shape of the energy distribution. The probability that the system has energy in a window about $E$ is the number of microstates there times the Boltzmann factor,

with $g(E)$ the density of states. The two factors pull in opposite directions: $g(E)$ rises steeply with energy — like $E^{3N/2}$ for an ideal gas — while $e^{-\beta E}$ falls. Their product is sharply peaked at the energy $E^{\ast }$ where the two rates balance, $\partial \ln g/\partial E=\beta$ — the same condition $1/T=\partial S/\partial E$ that fixes the microcanonical energy.⁴

Expanding $\ln P(E)$ about the peak gives a Gaussian,

of width $\sqrt{⟨\Delta E^2⟩}=\sqrt{k_B{T}^2{C}_V}\sim \sqrt{N}$. The peak position $E^{\ast }=U\sim N$ grows faster than the width, so the fractional width $\sqrt{N}/N=N^{-1/2}$ vanishes in the thermodynamic limit. The canonical energy distribution is a spike at $U$ with a relative width of order $N^{-1/2}$.

Equivalence of the canonical and microcanonical ensembles

The two ensembles describe the same physics because the canonical energy distribution is concentrated at a single energy. In the microcanonical ensemble the energy is fixed at $E$ and the entropy is $S(E)=k_B\ln g(E)$. In the canonical ensemble the energy is distributed as $P(E)$, sharply peaked at $E^{\ast }=U$; evaluating any thermodynamic quantity as a canonical average is the same as evaluating it at the single energy $E^{\ast }$, because the distribution has negligible weight anywhere else.⁵

The correspondence can be made precise through the free energy. Writing $P(E)\propto e^{[S(E)/k_B]-\beta E}$, the peak maximizes $S(E)-E/T$, so the minimum of $E-TS(E)$ over $E$ is the Helmholtz free energy,

This is the Legendre transform connecting the microcanonical entropy $S(E)$ to the canonical free energy $F(T)$. The transform is exact in the thermodynamic limit precisely because the distribution is sharp: the saddle-point (Laplace) evaluation of the sum $Z={\sum }_{E}g(E)e^{-\beta E}$ is dominated by $E^{\ast }$ with corrections of relative order $N^{-1/2}$.

Breakdown at phase transitions

The equivalence rests on the energy distribution having a single sharp peak. Where that fails, the ensembles can disagree. At a first-order phase transition the density of states develops structure such that $P(E)$ becomes bimodal — two peaks at the energies of the coexisting phases, separated by a suppressed region. The canonical ensemble at the transition temperature averages over both peaks, giving an energy intermediate between the two phases, while the microcanonical ensemble at an energy between the peaks describes a genuine two-phase mixture with its own properties. The heat capacity, which measures the width, diverges as the two peaks merge at a continuous transition. These are the cases where the choice of ensemble matters, and they are taken up in the phase-transition module; away from them, and for any system with short-range interactions in the thermodynamic limit, the canonical and microcanonical descriptions coincide.⁶

Summary

• The energy variance in the canonical ensemble is the second logarithmic derivative of the partition function, $⟨\Delta E^2⟩={\partial }^2\ln Z/\partial {\beta }^2=-\partial U/\partial \beta$; $\ln Z$ generates the energy cumulants.
• The fluctuation–response identity $⟨\Delta E^2⟩=k_B{T}^2{C}_V$ ties the equilibrium energy spread to the heat capacity and forces $C_V\ge 0$.
• The relative fluctuation scales as $1/\sqrt{N}$, of order $10^{-12}$ for a mole, so the canonical energy distribution is a Gaussian spike at $U$ with relative width $N^{-1/2}$, peaked where $\partial \ln g/\partial E=\beta$.
• Because the peak is sharp, the canonical and microcanonical ensembles give the same intensive thermodynamics as $N\to \infty$, related by the Legendre transform $F(T)={\min }_E[E-TS(E)]$; the equivalence breaks only where $P(E)$ loses its single sharp peak, at phase transitions.