If we change the temperature of a given system, there will be a relation between its entropy and temperature S(T). Is S(T) the same in a canonical ensemble and a grand canonical ensemble? If not, is there any example in which S(T) is qualitatively different in different ensembles?
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2 Answers
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There is a very nice property that works, in practice, for most systems that is that of equivalence between the Gibbs' ensembles in the thermodynamic limit. The prototypical example is that of the equivalence between the canonical ensemble and the microcanonical ensemble.
One way to state it is to say that the free energy $F(T,N,V) \equiv -k_BT \ln Q(T,N,V) = \langle E \rangle_T -TS_{c}(T,N,V)$ (where $Q(T,N,V)$ is the partition function and $S_c(T,N,V)$ the canonical entropy) will tend, in the thermodynamic limit, towards $\langle E\rangle_{T} - TS_{m}(\langle E\rangle_T)$, where $S_m(E) = k_B \ln \Omega(E,N,V)$ is the microcanonical entropy.
In this context, $\langle E\rangle_{T}$ is a function (possibly complicated) of the temperature $T$. But what the equivalence between ensembles tells us is that:
$S_c(T,N,V) \sim S_m(\langle E\rangle_{T}, N,V)$
If now, instead of $\langle E\rangle_{T}$ we write explicitly the functional dependence as $\langle E\rangle_{T} = \phi(T,N,V)$, we find that
$S_c(T,N,V) \sim S_m(\phi(T,N,V), N,V)$ in the thermodynamic limit.
For this result not to be a contradiction, it has to be the case that the functional dependence of $S_c$ in $T$ has to be the same as that of $S_m$.
Two points are in order here:
This reply is very similar to that of Tom-Tom but here relies explicitly on the equivalence property between ensemble in the thermodynamic limit. As a consequence, it is not true for finite system sizes in general.
This question only arises in statistical thermodynamics. In traditional thermodynamics (without accounting for equilibrium fluctuations), there is one entropy that is a function of state and there exist equations relating state variables with one another (most of the time using Schwartz' theorem).
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The statistical ensembles differ in the constraints imposed to them. In the canonical ensemble, the number of particles $N$, the volume $V$ and the temperature $T$ are fixed. In the grand-canonical ensemble, the number of particles is not fixed, it is determined by the chemical potential $\mu$, which plays the same role on $N$ as temperature on energy or pressure on the volume.
At equilibrium, the state functions, such as $S$, depend on these imposed constraints: we write $S(N,V,T)$ for the entropy in the canonical ensemble, and $S(\mu,V,T)$ in the grand-canonical ensemble.
If in a grand-canonical ensemble the average number of particles is $\langle N\rangle$, determined by $\mu$, then the average entropy $\langle S\rangle_{\text{grand canonical}}(\mu,V,T)$ is defined to equal $S_{\text{canonical}}(\langle N\rangle, V,T)$. This is the only consistent way to define the grand-canonical entropy, justified by the fact that the grand-canonical ensemble is a weighted set of replicas of the canonical ensemble with different values of $N$, which average is imposed by $\mu$.
