Energy and Density Fluctuation

1. Canonical Ensemble

A canonical ensemble represents a system:

In thermal equilibrium with a heat reservoir
At fixed temperature $T$
Fixed volume $V$
Fixed number of particles $N$

The system can exchange energy with the reservoir.

2. Probability of a Microstate

If the system has discrete energy levels $E_i$, the probability of finding the system in the $i^{th}$ microstate is:

\[ P_i = \frac{e^{-\beta E_i}}{Z} \]

where

\[ \beta = \frac{1}{k_B T} \]

3. Canonical Partition Function

The canonical partition function is defined as:

\[ Z = \sum_i e^{-\beta E_i} \]

For continuous energy spectrum:

\[ Z = \int e^{-\beta E} g(E) \, dE \]

where $g(E)$ is the density of states.

4. Importance of Partition Function

The partition function is the central quantity of statistical mechanics. All thermodynamic properties can be derived from it.

Some important relations:

4.1 Helmholtz Free Energy

\[ F = -k_B T \ln Z \]

4.2 Mean Energy

\[ \langle E \rangle = - \frac{\partial}{\partial \beta} \ln Z \]

4.3 Entropy

\[ S = -\left( \frac{\partial F}{\partial T} \right)_V \]

5. Energy Fluctuation

Since the system exchanges energy with the reservoir, the energy is not fixed. Therefore, there are fluctuations in energy.

5.1 Mean Square Energy

\[ \langle E^2 \rangle = \frac{1}{Z} \sum_i E_i^2 e^{-\beta E_i} \]

5.2 Energy Fluctuation

The fluctuation in energy is given by:

\[ (\Delta E)^2 = \langle E^2 \rangle - \langle E \rangle^2 \]

Proof of Fluctuations in Energy in Canonical Ensemble

Mean energy of the ensemble is:

$$ \begin{aligned} \langle E \rangle &= \dfrac{\sum n_i E_i}{\sum n_i} \\ &= \dfrac{\sum \dfrac{N}{Z} g_i e^{-\dfrac{E_i}{kT}} E_i}{\sum \dfrac{N}{Z} g_i e^{-\dfrac{E_i}{kT}}} \\ &= \dfrac{\sum g_i e^{-\dfrac{E_i}{kT}} E_i}{\sum g_i e^{-\dfrac{E_i}{kT}}}\\ &= \dfrac{\sum g_n E_n e^{-\dfrac{E_n}{kT}}}{\sum g_n e^{-\dfrac{E_n}{kT}}} \end{aligned} $$

Now differentiating with respect to $ T $:

\[ \begin{aligned} \dfrac{\partial}{\partial T} \langle E \rangle &= \dfrac{ \sum g_n E_n e^{-\dfrac{E_n}{kT}} \left( \dfrac{E_n}{kT^2} \right) \left( \sum g_n e^{-\dfrac{E_n}{kT}} \right) - \left( \sum g_n e^{-\dfrac{E_n}{kT}} \dfrac{E_n}{kT^2} \right) \left( \sum g_n E_n e^{-\dfrac{E_n}{kT}} \right) }{ \left( \sum g_n e^{-\dfrac{E_n}{kT}} \right)^2 } \\ &= \dfrac{1}{kT^2} \left[ \dfrac{ \sum g_n E_n^2 e^{-\dfrac{E_n}{kT}} \left( \sum g_n e^{-\dfrac{E_n}{kT}} \right) - \left( \sum g_n E_n e^{-\dfrac{E_n}{kT}} \right)^2 }{ \left( \sum g_n e^{-\dfrac{E_n}{kT}} \right)^2 } \right] \\ &= \dfrac{1}{kT^2} \left[ \left\langle E_n^2 \right\rangle - \left\langle E_n \right\rangle^2 \right]\\ \Rightarrow kT^2 \dfrac{\partial}{\partial T} \langle E \rangle &= \left\langle (\Delta E)^2 \right\rangle\\ \Rightarrow \left\langle (\Delta E)^2 \right\rangle &= kT^2 C_V \end{aligned} \]

Relative RMS fluctuation in the energy with respect to the mean energy is:

\[ \begin{aligned} \dfrac{\sqrt{ \left\langle (\Delta E)^2 \right\rangle }}{ \langle E \rangle } &= \dfrac{ \sqrt{ kT^2 C_V } }{ \langle E \rangle } \\ &= \dfrac{ \sqrt{ kT^2 Nk } }{ NkT } \\ &= N^{-1/2} \\ &= \left(10^{23}\right)^{-1/2} \\ &= 10^{-11} \end{aligned} \]

Which is very small. Hence, in a canonical ensemble, the energy is almost equal to the mean energy.

6. Relation Between Energy Fluctuation and Heat Capacity

Using statistical relations, one can show:

\[ (\Delta E)^2 = k_B T^2 C_V \]

where $C_V$ is the heat capacity at constant volume.

7. Relative Energy Fluctuation

The relative fluctuation is:

\[ \frac{\Delta E}{\langle E \rangle} \]

For macroscopic systems:

\[ \frac{\Delta E}{\langle E \rangle} \propto \frac{1}{\sqrt{N}} \]

As $N \to \infty$, relative energy fluctuation → 0.

Therefore, for large systems, the energy becomes sharply defined, and canonical ensemble predictions agree with thermodynamics.

8. Important Result

Energy fluctuations in the canonical ensemble are extremely small for macroscopic systems but become significant for small systems (nanoscopic systems).

Fluctuations in the Density of a Grand Canonical Ensemble

In a Grand Canonical Ensemble, the number of particles $N$ and energy $E$ both fluctuate. Since the density $\rho = \dfrac{N}{V}$ and the volume $V$ is fixed, fluctuations in density are equivalent to fluctuations in $N$.

The average number of particles is:

\[ \left\langle N \right\rangle = \dfrac{\sum_N \sum_n N e^{-\dfrac{E_{nN} - \mu N}{kT}}}{\sum_N \sum_n e^{-\dfrac{E_{nN} - \mu N}{kT}}} \]

\[ \Rightarrow \left( \dfrac{\partial \left\langle N \right\rangle}{\partial \mu} \right)_T = \dfrac{1}{kT} \left[ \dfrac{\sum_N \sum_n N^2 e^{-\dfrac{E_{nN} - \mu N}{kT}}}{\sum_N \sum_n e^{-\dfrac{E_{nN} - \mu N}{kT}}} - \left( \dfrac{\sum_N \sum_n N e^{-\dfrac{E_{nN} - \mu N}{kT}}}{\sum_N \sum_n e^{-\dfrac{E_{nN} - \mu N}{kT}}} \right)^2 \right] \]

\[\begin{aligned} &= \dfrac{1}{kT} \left( \left\langle N^2 \right\rangle - \left\langle N \right\rangle^2 \right) \\ &= \dfrac{(\Delta N)^2}{kT} \\ \Rightarrow \left\langle (\Delta N)^2 \right\rangle &= kT \left( \dfrac{\partial \left\langle N \right\rangle}{\partial \mu} \right)_T \end{aligned} \]

This gives the mean square fluctuation in the number of particles (or density).

Fluctuations in Energy of a Grand Canonical Ensemble

The mean energy of a grand canonical ensemble is:

\[ \left\langle E_{nN} \right\rangle = \dfrac{\sum g_n E_{nN} e^{-\dfrac{E_{nN} - \mu N}{kT}}}{\sum g_n e^{-\dfrac{E_{nN} - \mu N}{kT}}} \]

Now, differentiate with respect to $T$:

\[ \begin{aligned} \dfrac{\partial \left\langle E_{nN} \right\rangle}{\partial T} &= \dfrac{ \sum g_n E_{nN} e^{-\dfrac{E_{nN} - \mu N}{kT}} \cdot \dfrac{E_{nN} - \mu N}{kT^2} \cdot \sum g_n e^{-\dfrac{E_{nN} - \mu N}{kT}} - \sum g_n E_{nN} e^{-\dfrac{E_{nN} - \mu N}{kT}} \cdot \sum g_n \dfrac{E_{nN} - \mu N}{kT^2} e^{-\dfrac{E_{nN} - \mu N}{kT}} }{ \left( \sum g_n e^{-\dfrac{E_{nN} - \mu N}{kT}} \right)^2 } \\ &= \dfrac{1}{kT^2} \left[ \dfrac{ \sum g_n E_{nN}^2 e^{-\dfrac{E_{nN} - \mu N}{kT}} }{ \sum g_n e^{-\dfrac{E_{nN} - \mu N}{kT}} } - \left( \dfrac{ \sum g_n E_{nN} e^{-\dfrac{E_{nN} - \mu N}{kT}} }{ \sum g_n e^{-\dfrac{E_{nN} - \mu N}{kT}} } \right)^2 \right] \\ &= \dfrac{1}{kT^2} \left( \left\langle E_{nN}^2 \right\rangle - \left\langle E_{nN} \right\rangle^2 \right)\\ \Rightarrow \left\langle (\Delta E)^2 \right\rangle &= kT^2 \dfrac{\partial \left\langle E \right\rangle}{\partial T} \\ &= kT^2 C_V \end{aligned} \]

This represents the mean square fluctuation in energy in terms of the heat capacity at constant volume.