4.5: Energy Fluctuations in the Canonical Ensemble
- Page ID
- 5231
In the canonical ensemble, the total energy is not conserved. ( \(H (x) \ne \text {const} \) ). What are the fluctuations in the energy? The energy fluctuations are given by the root mean square deviation of the Hamiltonian from its average \(\langle H \rangle \):
\[ \Delta E = \sqrt{\langle\left(H-\langle H\rangle\right)^2\rangle} =\sqrt{\langle H^2 \rangle - \langle H \rangle^2} \nonumber \]
\(\langle H \rangle = - \frac {\partial}{\partial \beta} \ln Q (N,V,T) \)
\(\langle H^2 \rangle = \frac {1}{Q} C_N \int dx H^2 (x) e^{- \beta H (x)} \) }}
\(= \frac{1}{Q} C_N \int dx \frac{\partial^2}{\partial \beta^2}e^{-\beta H(x)}\)
\(= \frac{1}{Q} \frac {\partial^2}{\partial \beta^2}Q\)
\(= \frac{\partial^2}{\partial \beta^2}\ln Q + \frac {1}{Q^2} \left( \frac {\partial Q}{\partial \beta}\right)^2\)
\(= \frac{\partial^2}{\partial \beta^2}\ln Q +\left[\frac{1}{Q} \frac{\partial Q}{\partial \beta}\right]^2\)
\(= \frac{\partial^2}{\partial \beta^2}\ln Q + \left[ \frac {\partial}{\partial \beta}\ln Q\right]^2\)
Therefore
\[\langle H^2 \rangle - \langle H\rangle^2 =\frac{\partial^2}{\partial \beta^2}\ln Q \nonumber \]
But
\[ \frac{\partial^2}{\partial \beta^2}\ln Q = kT^2 C_V \nonumber \]
Thus,
\[\Delta E = \sqrt{kT^2 C_V} \nonumber \]
Therefore, the relative energy fluctuation \(\frac {\Delta E}{E} \) is given by
\[\frac{\Delta E}{E} = \frac{\sqrt{kT^2 C_V}}{E} \nonumber \]
Now consider what happens when the system is taken to be very large. In fact, we will define a formal limit called the thermodynamic limit, in which \(N\longrightarrow\infty\) and \(V\longrightarrow\infty\) such that \(\frac {N}{V} \) remains constant.
Since \(C_V\) and \(E\) are both extensive variables, \(C_V\sim N\) and \(E \sim N\),
\[\frac {\Delta E}{E} \sim \frac{1}{\sqrt{N}} \longrightarrow 0\;\;\;{as}\;\;\;N\rightarrow \infty \nonumber \]
But \(\frac {\Delta E}{E} \) would be exactly 0 in the microcanonical ensemble. Thus, in the thermodynamic limit, the canonical and microcanonical ensembles are equivalent, since the energy fluctuations become vanishingly small.