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{ \Omega(T, V, \mu) \equiv - k T \ln{\mathcal{Z}} = \Expval{U_A} - T S_A - \mu \Expval{N_A} } \end{aligned}

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

References

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