||Message]
Category:
The entropy as computed from classical statistical mechanics does not meet the requirement that it should be an extensive property. This is known as Gibbs paradox. This is resolved by noting that the correct counting of micro states must satisfy the requirement imposed by quantum mechanics on states of a system of identical particles.
For a system of \(N\) identical particles there are \(N!\) micro states that are obtained when under different permutations of particles.In classical physics these states are treated as different micro states but not in quantum mechanics.
As example in a system of two identical particles we can think of two micro states:
- In the first micro state, MS1, with first particle has position \({\vec r}_1\) and the second particle has position \({\vec r}_2\).
- Next consider a micro state, MS2, obtained from the micro state MS1 by exchange of the two particles. Thus in the second micro state MS2, first particle has position \({\vec r}_2\) and the second particle has position \({\vec r}_2\).
In classical physics these two micro states , MS1 and MS2, are treated as different. However quantum theory demands, that a system of identical particles, these two micro states should not be counted as being different.
Similar statement applies to system of \(N\) identical particles. The classical counting of micro states leads to a number \(\Omega_\text{cl}\) that is \(N!\) times the number, \(\Omega_{qm}\) as per counting in quantum mechanics. The canonical partition function is given by
\begin{equation}
Z = \sum_E \Omega{E} e^{-\beta E}
\end{equation}
This we get
\begin{equation}
Z_{\text{qm}} = \frac{1}{N!} Z_{\text{cl}}
\end{equation}
It can now be checked that the expression for entropy as derived from \(Z_\text{qm}\) obeys the requirement of being extensive.
