The grand canonical ensemble

The grand canonical ensemble

The grand canonical ensemble is a statistical model that can be used to describe the behavior of a system with a constant temperature, a constant chemical potential and a constant volume/pressure. (In what follows we will enforce constant a volume.) If we have a Hamiltonian, $H(x,p)$, for the system the isothermal-isobaric partition function can be calculated using: $$ Z_{gc} = \sum_i e^{-\beta H(x_i,p_i) } e^{\beta \mu N(x_i,p_i) } $$ The sum here runs over all the microstates in the system and is replaced by an integral if you have a continuous rather than a discrete state space. $\beta$, meanwhile, is equal to one divided by the temperature, $T$, multiplied by Boltzmann's constant, $k_B$. That is to say $\beta = \frac{1}{k_B T}$. The probabibility of being with an individual microstate in the grand canonical ensemble is given by: $$ p_i = \frac{ e^{-\beta H(x_i,p_i)} e^{\beta \mu N(x_i,p_i) } }{ Z_i } $$ We can thus calculate the average number of atoms in the system by computing an emsemble average using: $$ \langle N \rangle = \sum_i N(x_i,p_i) p_i $$ or by using the analytical result below: $$ \langle N \rangle = \frac{ \partial \ln Z_{gc} }{\partial (\beta \mu) } $$ There is a similar result that relates the second derivative of the logarithm of the partition function with the fluctuations in the number of atoms: $$ \langle (N - \langle N \rangle)^2 \rangle = \left( \frac{\partial^2 \ln Z_{gc}(\mu,V,T)}{\partial (\beta \mu)^2} \right) $$

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