7.6: Fluctuations

We will now calculate the fluctuations in the values of energy and the number of particles as given by the canonical and grand canonical ensembles. First consider $N$. From the definition, we have

$$\begin{equation} \begin{split} \frac{1}{\beta} \frac{\partial Z}{\partial \mu} & = Z \langle N \rangle = Z\;\bar{N} \\[0.125in] \frac{1}{\beta^2} \frac{\partial^2 Z}{\partial \mu^2} & = Z \langle N^2 \rangle \\[0.125in] \end{split} \end{equation} \label{7.6.1}$$

If we calculate $\bar{N}$ from the partition function as a function of $β$ and $µ$, we can differentiate it with respect to $µ$ to get

$$\frac{1}{\beta} \frac{\partial \bar{N}}{\partial \mu} = -\frac{1}{Z^2 \beta^2} \left( \frac{\partial Z}{\partial \mu} \right)^2 + \frac{1}{\beta^2} \frac{\partial^2 Z}{\partial \mu^2} = \langle N^2 \rangle - \langle N \rangle^2 = \Delta N^2 \label{7.6.2}$$

The Gibbs free energy is given by $G = µ\bar{N}$ and it also obeys

$$dG = −SdT + V dp + µd\bar{N}$$

These follow from Equation 5.1.16 and Equation 5.1.10. Since $T$ is fixed in the differentiations we are considering, this gives

$$\frac{\partial µ}{\partial \bar{N}} = \frac{V}{\bar{N}} \frac{\partial p}{\partial \bar{N}} \label{7.6.4}$$

The equation of state gives p as a function of the number density $ρ ≡ \frac{\bar{N}}{V}$, at fixed temperature. Thus

$$\frac{\partial p}{\partial \bar{N}} = \frac{1}{V} \frac{\partial p}{\partial ρ}$$

Using this in Equation $\ref{7.6.4}$, we get

$$\frac{1}{\beta} \frac{\partial \bar{N}}{\partial \mu} = \bar{N} \frac{kT}{\frac{\partial p}{\partial ρ}}$$

From Equation $\ref{7.6.2}$, we now see that the mean square fluctuation in the number is given by

$$\frac{\Delta N^2}{N^2} = \frac{1}{\bar{N}} \frac{kT}{\frac{\partial p}{\partial ρ}}$$

This goes to zero as $\bar{N}$ becomes large, in the thermodynamic limit. An exception could occur if $\left(\frac{∂p}{∂ρ}\right)$ becomes very small. This can happen at a second order phase transition point. The result is that fluctuations in numbers become very large at the transition. The theoretical treatment of such a situation needs more specialized techniques.

We now turn to energy fluctuations in the canonical ensemble. For this we consider $N$ to be fixed and write

$$\begin{equation} \begin{split} \frac{\partial U}{\partial \beta} & = \frac{\partial}{\partial \beta} \left[ -\frac{1}{Q_N} \frac{\partial Q_N}{\partial \beta} \right] = \left[\frac{1}{Q_N} \frac{\partial Q_N}{\partial \beta} \right]^2 - \left[\frac{1}{Q_N} \frac{\partial^2 Q_N}{\partial \beta^2} \right] \\[0.125in] & = \langle H \rangle^2 - \langle H^2 \rangle \equiv -\Delta U^2 \\[0.125in] \end{split} \end{equation}$$

The derivative of $U = \langle H \rangle$ with respect to $\bar{T}$ gives the specific heat, so we find

$$\Delta U^2 = k C_v T^2,\;\;\; \frac{\Delta U^2}{U^2} = \frac{k C_v T^2}{U^2} ∼ \frac{1}{N}$$

Once again, the fluctuations are small compared to the average value as $N$ becomes large.