Grand Canonical Ensemble

So, far we have considered systems which are isolated (either completely or at least mechanically). Let us now consider more realistic systems which are in neither mechanically nor thermally isolated. In other words, in this section we shall deal with systems which are in thermal contact with a heat bath (so that the temperature is kept constant) and also exchanges particles with the environment (with the chemical potential kept constant). Therefore, now the $\Gamma$-space comprises of all the canonical momenta and co-ordinates of systems with 0, 1, 2, .., $N$ particles. And the phase-space density $\rho$ is a function not only of ($p,q$) but of $N$ as well.

Consider a mechanically isolated system in contact with a heat bath (so that the system can be treated using Canonical Ensemble Approach) defined by $N, V, T$. Consider two subsystems of this which can exchange energy, as well as particles defined by $N_1, V_1, T$ and $N_2, V_2, T$. Then,

$$\rho(p_1, q_1, N_1) \propto \mbox{probability of $N_1$ particles being within volume $V_1$ with co-ordinates $(p_1, q_1)$.}$$

Let us assume that $V_1 \ll V$ and $N_1 \ll N$. Then, neglecting the inter-particle interactions across the surface separating the two subsystems, we have,

$$\rho(p_1, q_1, N_1) \propto e^{-\beta H(p_1, q_1, N_1)} \int_{V_2} dp_2 dq_2 e^{-\beta H(p_2, q_2, N_2)},$$ (47)

where, $H_1$ and $H_2$ have the same functional form. Choosing appropriate proportionality constants, we have,

$$\rho(p_1, q_1, N_1) = \frac{N!}{N_1! N_2!} \ \frac{e^{-\beta H(p_1, q_1, N_1)} \in... ...dq_2 e^{-\beta H(p_2 , q_2, N_2)}} {\int_V dp dq e^{-\beta H(p, q, N)}}$$

$$= \frac{Q_{N_2} (V_2, T )}{Q_N(V, T)} \frac{e^{-\beta H(p_1, q_1, N_1)}}{N_1! h^{3 N_1}},$$ (48)

so that,

$$\sum \limits^N _{N_1= 0} \int dp_1 dq_1 \rho (p_1, q_1, N_1 ) = 1.$$ (49)

Now,

$$Q_{N_2}(V_2 , T) / Q_N(V, T) = e^{- \beta \left[A(N_2, V_2, T) - A(N, V, T) \right.}$$

and,

$$A(N_2, V_2, T) - A(N, V, T) = A(N-N_1, V-V_1, T) - A(N,V,T)... ..._1 \mu + V_1 P \mbox{for} N_1, V_1 \ll N,V. \nonumber$$

where,

$$\mu = \frac{\partial A (N_2, V, T) }{\partial N_2} \Large \vert _{N_2 = N},$$ (50)

$$P = - \frac{\partial A (N, V_2, T)}{\partial V_2} \Large \vert _{V_2 = V}.$$ (51)

We define the Grand Partition Function to be,

$${\cal Z}(z,V,T) = \sum \limits_{N= 0}^\infty z^N Q_N (V, T),$$ (52)

where, $z = e^{\beta \mu}$ and $Q_N$ is the Partition function of a system with $N$ particles. Since,

$$\frac{Q_{N_2}(V_2, T)}{Q_N(V, T)} = z^{N_1} e^{-\beta V_1 P }$$

we have,

$$\rho(p, q, N) = \frac{z^N}{N! h^{3N}} e^{-\beta PV - \beta H(p, q)}.$$ (53)

The recipe for thermodynamics is given by      . and,

$$U = - \frac{\partial}{\partial \beta} \ln {\cal Z}(z, V, T).$$ (54)

Alternatively,

$$A= Nk_{\rm B}T \ln z - k_{\rm B} T \ln {\cal Z}(z, V, T).$$ (55)

The ensemble average of the particle number is given by,

$$\langle N \rangle = \frac{\sum \limits^\infty_{N=0} N' z^... ...(V, T)} = z \frac{\partial }{\partial z} \ln {\cal Z} (z,V,T)$$ (56)

Similarly, the mean square fluctuation of the number of particles is given by,

$$\langle N^2 \rangle - \langle N \rangle^2 = z^2 \frac{\partial^2 }{\partial z^2} \ln {\cal Z} (z,V,T)$$ (57)