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.