Fluctuations in Canonical and Grand Canonical Ensembles

1. Energy Fluctuations in the Canonical Ensemble

Let's begin with a system at constant $(T,V,N)$. The energy $E$ is not fixed; it fluctuates as the system exchanges energy with the heat bath. We want to quantify the size of these fluctuations. The standard measure is the variance, or mean-square fluctuation, defined as:
$⟨(\Delta E{)}^2⟩\equiv ⟨(E-⟨E⟩{)}^2⟩=⟨E^2⟩-⟨E{⟩}^2$
We already have an expression for the mean energy, $⟨E⟩=-\frac{\partial \ln Z}{\partial \beta }$. Now we need a way to find $⟨E^2⟩$. Recall the definition of the canonical average:
$⟨E^2⟩={\sum }_j{E}_j^2{P}_j=\frac{1}{Z}{\sum }_j{E}_j^2{e}^{-\beta E_j}$
Notice that the sum ${\sum }_j{E}_j^2{e}^{-\beta E_j}$ can be generated by differentiating the partition function $Z={\sum }_j{e}^{-\beta E_j}$ twice with respect to $\beta$:
$\frac{{\partial }^{2}Z}{\partial {\beta }^2}=\frac{\partial }{\partial \beta }\left({\sum }_j(-E_j)e^{-\beta E_j}\right)={\sum }_j(-E_j{)}^2{e}^{-\beta E_j}={\sum }_j{E}_j^2{e}^{-\beta E_j}$
Therefore, the mean-square energy is simply $⟨E^2⟩=\frac{1}{Z}\frac{{\partial }^{2}Z}{\partial {\beta }^2}$.

Assembling the variance gives $⟨(\Delta E{)}^2⟩=\frac{1}{Z}\frac{{\partial }^{2}Z}{\partial {\beta }^2}-{\left(-\frac{1}{Z}\frac{\partial Z}{\partial \beta }\right)}^2$. This expression can be simplified considerably. Recognizing it as the result of the quotient rule, we can see it is the second derivative of $\ln Z$:
$\frac{{\partial }^2(\ln Z)}{\partial {\beta }^2}=\frac{\partial }{\partial \beta }\left(\frac{1}{Z}\frac{\partial Z}{\partial \beta }\right)=\frac{1}{Z}\frac{{\partial }^{2}Z}{\partial {\beta }^2}-\frac{1}{Z^2}{\left(\frac{\partial Z}{\partial \beta }\right)}^2=⟨E^2⟩-⟨E{⟩}^2$
Thus, we arrive at the elegant and compact result:
$⟨(\Delta E{)}^2⟩=\frac{{\partial }^2\ln Z}{\partial {\beta }^2}$
Since $⟨E⟩=-(\partial \ln Z/\partial \beta )$, we can also write this as:
$⟨(\Delta E{)}^2⟩=-\frac{\partial ⟨E⟩}{\partial \beta }$

The following resources provide detailed derivations of this fundamental result.

2. Connection to Heat Capacity

The relation $⟨(\Delta E{)}^2⟩=-\partial ⟨E⟩/\partial \beta$ is powerful, but its physical meaning becomes clearer when we connect it to a measurable thermodynamic quantity: the heat capacity at constant volume, $C_V$.
$C_V={\left(\frac{\partial ⟨E⟩}{\partial T}\right)}_V$
Using the chain rule with $\beta =1/(k_{B}T)$, we have $\frac{\partial }{\partial \beta }=\frac{dT}{d\beta }\frac{\partial }{\partial T}=(-k_B{T}^2)\frac{\partial }{\partial T}$. Therefore:
${\left(\frac{\partial ⟨E⟩}{\partial \beta }\right)}_V=-k_B{T}^2{\left(\frac{\partial ⟨E⟩}{\partial T}\right)}_V=-k_B{T}^2{C}_V$
Substituting this into our fluctuation formula gives the celebrated result:
$⟨(\Delta E{)}^2⟩=-(-k_B{T}^2{C}_V)=k_B{T}^2{C}_V$
This is a profound connection. It states that the magnitude of energy fluctuations in a system at thermal equilibrium is directly proportional to its heat capacity. A system with a large heat capacity can absorb a lot of energy for a small change in temperature, and our result shows this is microscopically associated with large equilibrium fluctuations in its energy. Conversely, a system with a small $C_V$ has very small energy fluctuations. This provides a direct statistical mechanical interpretation of a purely thermodynamic quantity.

The following resources walk through this derivation and its implications.

3. Particle Number Fluctuations in the Grand Canonical Ensemble

The logic for deriving particle number fluctuations in the grand canonical ensemble $(T,V,\mu )$ is perfectly analogous. Here, the number of particles $N$ fluctuates as the system exchanges particles with a reservoir. The variance is:
$⟨(\Delta N{)}^2⟩=⟨N^2⟩-⟨N{⟩}^2$
From the previous lesson, we know $⟨N⟩=\frac{1}{\beta }\frac{\partial \ln \Xi }{\partial \mu }$.
The mean-square particle number $⟨N^2⟩$ can be obtained by differentiating the grand partition function $\Xi ={\sum }_{j,N}{e}^{-\beta (E_j-\mu N)}$ twice with respect to $\mu$:
$\frac{{\partial }^2\Xi }{\partial {\mu }^2}={\sum }_{j,N}(\beta N{)}^2{e}^{-\beta (E_j-\mu N)}$
This leads to $⟨N^2⟩=\frac{1}{\Xi {\beta }^2}\frac{{\partial }^2\Xi }{\partial {\mu }^2}$.

Just as before, we can show that the full expression for the variance simplifies to the second derivative of the logarithm of the partition function:
$⟨(\Delta N{)}^2⟩=\frac{1}{{\beta }^2}\frac{{\partial }^2\ln \Xi }{\partial {\mu }^2}$
Since $⟨N⟩=\frac{1}{\beta }\frac{\partial \ln \Xi }{\partial \mu }$, we can also write this in terms of the derivative of the average particle number:
$⟨(\Delta N{)}^2⟩=\frac{1}{\beta }{\left(\frac{\partial ⟨N⟩}{\partial \mu }\right)}_{T,V}=k_{B}T{\left(\frac{\partial ⟨N⟩}{\partial \mu }\right)}_{T,V}$
The derivative $(\partial ⟨N⟩/\partial \mu {)}_{T,V}$ is related to the isothermal compressibility ${\kappa }_T$ of the system (${\kappa }_T=\frac{V}{⟨N{⟩}^2}(\frac{\partial ⟨N⟩}{\partial \mu }{)}_{T,V}$), so once again a fluctuation is related to a thermodynamic response function.

The following resources provide clear derivations of this result.

4. The Thermodynamic Limit and Ensemble Equivalence

These fluctuation formulae are the key to understanding why the different statistical ensembles give identical thermodynamic predictions for macroscopic systems.

Consider the relative fluctuation in energy: the ratio of the root-mean-square (RMS) fluctuation to the mean energy.
$\frac{\Delta E_{rms}}{⟨E⟩}=\frac{\sqrt{⟨(\Delta E{)}^2⟩}}{⟨E⟩}=\frac{\sqrt{k_B{T}^2{C}_V}}{⟨E⟩}$
Both the internal energy $⟨E⟩$ and the heat capacity $C_V$ are extensive properties, meaning they are proportional to the number of particles $N$ in the system. Thus, $⟨E⟩\propto N$ and $C_V\propto N$. Substituting this scaling into the ratio gives:
$\frac{\Delta E_{rms}}{⟨E⟩}\propto \frac{\sqrt{N}}{N}=\frac{1}{\sqrt{N}}$
Similarly, for particle number fluctuations, both $⟨N⟩$ and $(\partial ⟨N⟩/\partial \mu )$ are extensive, so $⟨(\Delta N{)}^2⟩\propto N$. The relative fluctuation in particle number therefore also scales as:
$\frac{\Delta N_{rms}}{⟨N⟩}\propto \frac{\sqrt{N}}{N}=\frac{1}{\sqrt{N}}$
In the thermodynamic limit ($N\to \infty$), the relative fluctuations of both energy and particle number vanish. For a macroscopic system where $N\sim 10^{23}$, the relative fluctuation is on the order of $10^{-11.5}$, which is experimentally undetectable.

This is the justification for the equivalence of ensembles. Although a system in the canonical ensemble can have any energy, its probability distribution is so sharply peaked around the mean value that for a macroscopic system, the energy is effectively fixed. Thus, it behaves identically to a system in the microcanonical ensemble. Likewise, a macroscopic system in the grand canonical ensemble has a particle number that is so sharply defined that it behaves identically to a system in the canonical ensemble. This is a crucial result, as it allows us to choose whichever ensemble is most mathematically convenient for a given problem.

The "Fluctuations and Response" reading and the NPTEL video on the Grand Canonical Ensemble both provide excellent discussions of this point.

Conclusion

In this lesson, we have extended our statistical mechanics toolkit by moving from first to second derivatives of the partition function. This has uncovered a deep connection between microscopic fluctuations and macroscopic response functions.

Key Takeaways:

• Second derivatives of the logarithm of the partition function correspond to the variance (mean-square fluctuation) of the conjugate thermodynamic variable.
• The energy fluctuation in the canonical ensemble is related to the heat capacity: $⟨(\Delta E{)}^2⟩=k_B{T}^2{C}_V$.
• The particle number fluctuation in the grand canonical ensemble is related to the derivative of the average particle number with respect to the chemical potential: $⟨(\Delta N{)}^2⟩=k_{B}T(\partial ⟨N⟩/\partial \mu {)}_{T,V}$.
• Relative fluctuations of extensive quantities scale as $1/\sqrt{N}$, vanishing in the thermodynamic limit. This provides the formal justification for the equivalence of statistical ensembles.

These fluctuation-response relations are elementary examples of a broader principle in statistical physics known as the Fluctuation-Dissipation Theorem. In our next lesson, we will apply the canonical ensemble formalism to derive another central result for classical systems: the equipartition theorem, which provides a simple but powerful rule for distributing energy among the degrees of freedom of a system.