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Next: 4.4 Problems for Chapter Up: 4. Statistical Thermodynamics Previous: 4.2 Canonical ensemble

4.3 Grand canonical ensemble

Once again we put a small system ($1$) in contact with a large one ($2$). However, this time we do not only permit the exchange of energy but also the crossing over of particles from one subsystem to the other.

Figure 4.3: System in contact with an energy and particle reservoir: $\rightarrow $ grand canonical ensemble

And as before we can write down the probability density in the phase space of the smaller system; it depends now both on the number of particles $N_{1}$ and on $\{ \vec{r}_{i}, \vec{v}_{i}; i=1, \dots N_{1} \}$, as follows:
p(\vec{r}, \vec{v}; N_{1}) \propto e^{\mu N_{1}/kT} e^{-E(\vec{r}, \vec{v})/kT}
\end{displaymath} (4.49)

Summing this density over all possible values of $N_{1}$ and integrating - at each $N_{1}$ - over all $\{ \vec{r}_{i}, \vec{v}_{i}; i=1, \dots N_{1} \}$ we obtain the grand partition function
Z(\mu,V_{1},T) \equiv \sum_{N_{1}=0}^{\infty} e^{N_{1}\mu/kT}
\end{displaymath} (4.50)

Its value is just the ``total statistical weight'' of all possible states of system $1$. Above all, it serves as the source function of thermodynamics.

From the grand partition function we can easily derive expressions for the various thermodynamic observables. For instance, putting $z \equiv e^{\mu /kT}$ and $\beta \equiv 1/kT$ we find
$\displaystyle P$ $\textstyle =$ $\displaystyle \frac{kT}{V} \ln Z(z,V,T)$ (4.51)
$\displaystyle N (\equiv \langle N \rangle)$ $\textstyle =$ $\displaystyle z \frac{\partial}{\partial z}
\ln Z(z,V,T) = kT \frac{\partial \ln Z}{\partial \mu}$ (4.52)
$\displaystyle U (\equiv \langle E \rangle)$ $\textstyle =$ $\displaystyle - \frac{\partial}{\partial \beta}
\ln Z(z,V,T) = kT^{2} \frac{\partial \ln Z}{\partial T}$ (4.53)

As a rule the - permitted - fluctuations of the number of particles remain small; in particular we have $\Delta N / N \approx 1/\sqrt{N}$. Thus the grand ensemble is again equivalent to others ensembles of statistical mechanics.

[To do: applet with MD simulation, averages taken only over particles in a partial volume -> same results!]

Example: Let us visit the ideal gas again. For the grand partition function we have

$\displaystyle Z(z,V,T)$ $\textstyle =$ $\displaystyle \sum_{N}z^{N}\frac{V^{N}}{N!}
\left( \frac{2 \pi m k T}{h^{2}}\right)^{3N/2}$  
  $\textstyle =$ $\displaystyle \sum_{N} \frac{y^{N}}{N!}\;\;{\rm with}\;
y \equiv Vz \left( \frac{2 \pi m k T}{h^{2}}\right)^{3/2}$ (4.54)


Z=\exp \left[-zV \left( \frac{2 \pi m k T}{h^{2}}\right)^{3/...
\ln Z = -zV \left( \frac{2 \pi m k T}{h^{2}}\right)^{3/2}
\end{displaymath} (4.55)

Using the formulae for internal energy and pressure we find

P=-kTz \left( \frac{2 \pi m k T}{h^{2}}\right)^{3/2}
...-kTz \frac{3V}{2}\left( \frac{2 \pi m k T}{h^{2}}\right)^{3/2}
\end{displaymath} (4.56)

Consequently, $P = 2U/3V$ or

P=\frac{2}{3V} \frac{3NkT}{2} = \frac{N}{V} kT
\end{displaymath} (4.57)

in keeping with the phenomenological ideal gas equation.

The states within the grand ensemble may again be sampled in a random manner. Just as in the canonical Monte Carlo procedure we produce a sequence of microstates $\vec{\Gamma}_{c}(m),   m=1, \dots M$ with the appropriate relative frequencies. The only change is that we now vary also the number of particles by occasionally adding or removing a particle. The stochastic rule for these insertions and removals is such that it agrees with the thermodynamic probability of such processes. By averaging some quantity over the ``Markov chain'' of configurations we again obtain an estimate of the respective thermodynamic observable.
next up previous
Next: 4.4 Problems for Chapter Up: 4. Statistical Thermodynamics Previous: 4.2 Canonical ensemble
Franz Vesely