Categories: Physics, Thermodynamic ensembles, Thermodynamics.

Grand canonical ensemble

The grand canonical ensemble or μVT ensemble extends the canonical ensemble by allowing the exchange of both energy \(U\) and particles \(N\) with an external reservoir, so that the conserved state functions are the temperature \(T\), the volume \(V\), and the chemical potential \(\mu\).

The derivation is practically identical to that of the canonical ensemble. We refer to the system of interest as \(A\), and the reservoir as \(B\). In total, \(A\!+\!B\) has energy \(U\) and population \(N\).

Let \(c_B(U_B)\) be the number of \(B\)-microstates with energy \(U_B\). Then the probability that \(A\) is in a specific microstate \(s_A\) is as follows:

\[\begin{aligned} p(s) = \frac{c_B\big(U - U_A(s_A), N - N_A(s_A)\big)}{\sum_{s_A} c_B\big(U \!-\! U_A(s_A), N \!-\! N_A(s_A)\big)} \end{aligned}\]

Then, as for the canonical ensemble, we assume \(U_B \gg U_A\) and \(N_B \gg N_A\), and approximate \(\ln{p(s_A)}\) by Taylor-expanding \(\ln{c_B}\) around \(U_B = U\) and \(N_B = N\). The resulting probability distribution is known as the Gibbs distribution, with \(\beta \equiv 1/(kT)\):

\[\begin{aligned} \boxed{ p(s_A) = \frac{1}{\mathcal{Z}} \exp\!\Big(\!-\! \beta \: \big( U_A(s_A) \!-\! \mu N_A(s_A) \big) \Big) } \end{aligned}\]

Where the normalizing grand partition function \(\mathcal{Z}(\mu, V, T)\) is defined as follows:

\[\begin{aligned} \boxed{ \mathcal{Z} \equiv \sum_{s_A}^{} \exp\!\Big(\!-\! \beta \: \big( U_A(s_A) - \mu N_A(s_A) \big) \Big) } \end{aligned}\]

In contrast to the canonical ensemble, whose thermodynamic potential was the Helmholtz free energy \(F\), the grand canonical ensemble instead minimizes the grand potential \(\Omega\):

\[\begin{aligned} \boxed{ \begin{aligned} \Omega(T, V, \mu) &\equiv - k T \ln{\mathcal{Z}} \\ &= \expval{U_A} - T S_A - \mu \expval{N_A} \end{aligned} } \end{aligned}\]

So \(\mathcal{Z} = \exp\!(- \beta \Omega)\). This is proven in the same way as for \(F\) in the canonical ensemble.

References

  1. H. Gould, J. Tobochnik, Statistical and thermal physics, 2nd edition, Princeton.

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