5.5 The classical approximation again

Take-home message: The Gibbs distribution gives an alternative way of treating the ideal gas in the classical limit.

The last time we calculated the properties of an ideal gas, we regarded the particle number as fixed, but the energy as variable (the temperature being fixed.) However we knew that for a large system, fluctuations are small, and the results will be essentially the same as the more difficult problem of an isolated gas with fixed energy.

This raises the question of whether we couldn't obtain the same results regarding the particle number as variable also, fixing instead the chemical potential. The answer is that we can of course, as the following shows.

From the previous sections on the Gibbs distribution and the ideal gas of bosons or fermions, we have

$$\begin{eqnarray*} \Phi_G\!\!\!&\equiv &\!\!\!-k_{\scriptscriptstyle B}T \ln{\cal... ...style B}T\sum_r\ln\left(1\pm e^{(\mu-\varepsilon_r)\beta}\right) \end{eqnarray*}$$

where $r$ labels the single particle energy levels, and the signs are for fermions and bosons respectively.

Now imagine that $e^{\mu\beta}\ll 1$, which requires $\mu$ to be large and negative. Never mind for a moment what that means physically. Then, using $\ln (1+x)\approx x$ for small $x$, we get

$$\begin{eqnarray*} \Phi_G\!\!\!&=&\!\!\!-k_{\scriptscriptstyle B}T\sum_r e^{\mu\b... ...\\ \!\!\!&=&\!\!\!-k_{\scriptscriptstyle B}T e^{\mu\beta}Z_1(T) \end{eqnarray*}$$

where $Z_1$ is the one-particle translational partition function (not grand p.f.) for an atom in an ideal gas. As we calculated previously, $Z_1(T)=V n_{\scriptscriptstyle Q}(T)$

From $\Phi_G$ we can find the average particle number:

$$\begin{eqnarray*} N\!\!\!&=&\!\!\!-\left({\partial \Phi_G\over\partial \mu}\right)_{\!\scriptstyle T,V} \\ \!\!\!&=&\!\!\!e^{\mu\beta}Z_1 \end{eqnarray*}$$

and solving for $\mu$ we get

$$\mu=-k_{\scriptscriptstyle B}T \ln(Z_1/N)=-k_{\scriptscriptstyle B}T \ln\left({n_{\scriptscriptstyle Q}\over n}\right)$$

So now we see that $\mu$ large and negative requires $n\ll n_{\scriptscriptstyle Q}$ or far fewer particles than states - exactly the classical limit as defined before.

Finally, since $\Phi_G=E-TS-\mu N = F-\mu N$, we have

$$\begin{eqnarray*} F\!\!\!&=&\!\!\!\Phi_G+\mu N \\ \!\!\!&=&\!\!\!-N k_{\scripts... ...!&=&\!\!\!-N k_{\scriptscriptstyle B}T \left(\ln(Z_1/N)+1\right) \end{eqnarray*}$$

However this is exactly what we get from $F=-k_{\scriptscriptstyle B}T\ln Z$ with $Z=(Z_1)^N/N !$. Thus we recover all our previous results.

References

• Mandl 11.4
• Bowley and Sánchez 10.2-3