Consider two noninteracting systems \(A\)
and \(B\). We can either treat these systems as separate, or as a
single combined system \(AB\). We can enumerate all states of the
combined by enumerating all states of each separate system. The
probability of the combined state \((i_A,j_B)\) is given by
\(P_{ij}^{AB} = P_i^AP_j^B\). In other words, the probabilities
combine in the same way as two dice rolls would, or the
probabilities of any other uncorrelated events.
- Show that the entropy of the combined system \(S_{AB}\) is the
sum of entropies of the two separate systems considered
individually, i.e. \(S_{AB} = S_A+S_B\). This means that entropy is
extensive. Use the Gibbs entropy for this computation. You need
make no approximation in solving this problem.
- Show that if you have \(N\) identical non-interacting systems,
their total entropy is \(NS_1\) where \(S_1\) is the entropy of a
single system.
Note
In real materials, we treat properties as being extensive even
when there are interactions in the system. In this case,
extensivity is a property of large systems, in which surface
effects may be neglected.