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 UU and particles NN with an external reservoir, so that the conserved state functions are the temperature TT, the volume VV, and the chemical potential μ\mu.

The derivation is practically identical to that of the canonical ensemble. We refer to the system of interest as AA, and the reservoir as BB. In total, A ⁣+ ⁣BA\!+\!B has energy UU and population NN.

Let cB(UB)c_B(U_B) be the number of BB-microstates with energy UBU_B. Then the probability that AA is in a specific microstate sAs_A is as follows:

p(s)=cB(UUA(sA),NNA(sA))sAcB(U ⁣ ⁣UA(sA),N ⁣ ⁣NA(sA))\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 UBUAU_B \gg U_A and NBNAN_B \gg N_A, and approximate lnp(sA)\ln{p(s_A)} by Taylor-expanding lncB\ln{c_B} around UB=UU_B = U and NB=NN_B = N. The resulting probability distribution is known as the Gibbs distribution, with β1/(kT)\beta \equiv 1/(kT):

p(sA)=1Zexp ⁣( ⁣ ⁣β(UA(sA) ⁣ ⁣μNA(sA)))\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 Z(μ,V,T)\mathcal{Z}(\mu, V, T) is defined as follows:

ZsAexp ⁣( ⁣ ⁣β(UA(sA)μNA(sA)))\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 FF, the grand canonical ensemble instead minimizes the grand potential Ω\Omega:

Ω(T,V,μ)kTlnZ=UATSAμNA\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 Z=exp(βΩ)\mathcal{Z} = \exp(- \beta \Omega). This is proven in the same way as for FF in the canonical ensemble.


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