In the chapter of "Thermal Physics, Blundell" that introduces calculating state functions from the partitions function of the system, takes into account a system that doesn't have degenerate energy levels and reaches to the usual results. Later, these results are applied on the system of ideal gas. However the ideal gas does have degenerate states. So, I tried to derive the furmulas that connect the partition function of a degenerate system and thermodynamic state functions and see if they are the same.

Calculation Of Entropy For A Degenerate System

Let's take for example the entropy. Since the system is degenerate, the probability of the system being in the state of energy $E_i$ is:

$$P_i = \frac{g_j e^{-βΕ_i}}{Z} $$ $$ ln {P_i}=-βΕ_i + ln{g_i}-ln{Z}$$

Now, the gibbs entropy is expressed as:

$$ S=-k_B \sum_{i} p_i ln{p_i}=$$ $$=-k_B \sum_{i} p_i (-βE_i + ln{g_i} - ln{Z})=$$ $$ = k_B \sum_{i} p_iE_i -k_B \sum_{i} p_i ln{g_i} + k_Bln{Z}$$

Therefore we get to the conclusion that:

$$ S= \frac{U}{T} + k_B \left<ln{g}\right> + k_BlnZ = \frac{U}{T} + k_B\left<ln{ \frac{Z}{g}}\right>$$

Application To Ideal Gas

Now, the density of states (degeneracy) for the ideal gas is: $$g(k)=\frac{Vk^2}{2π^2} $$

The energy levels $$E(k)=\frac{\hbar^2 k^2}{2m}$$

And The partition Function: $$Z_1 = \frac{1}{\hbar^3} \left(\frac{mk_BT}{2π} \right)^{3/2}$$

Now, the difference between the equations for the degenerate and the non-degenerate system is the term: $$ \left<ln{g_i} \right>= \int p_iln{g_i}=\frac{V}{2π^2Z} \int_{0}^{\infty} e^{-β\hbar^2k^2/2m}k^2ln{ \left(\frac{Vk^2}{2π^2}\right)}dk \neq 0$$

Comments On the Results

So the difference between the two cases in general in not zero. Therefore either my derivation is wrong or I haven't taken something into account...or both. Which is it?


The sum in the entropy formula is a sum over states not a sum over energy levels. In other words the degeneracy does not appear in the probabilities, however there will be multiple terms in the sum which corrispond to states with the same energy. This means there is no contribution to the logarithmic term from the degeneracy factor.

The degeneracy factor occurs when we group multiple states with the same energy into energy levels in order to simplify evaluating the sum.

In most places in statistical physics we are interested in expectations which are linear in the probabilities, so this distinction is largely a matter of notation. This can lead to people being slightly sloppy with phrasing. The Gibbs expression for the entropy is the only exception that springs to mind.

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