5

Albert Einstein's theory of thermal capacity of a solids makes the assumption that a crystal is made up from oscillators which of course oscillate, in all three directions. Thus, for N atoms of the crystal, we have 3N oscillators and each one is described by: $\ddot r = -\omega_E ^2 r $, where $r$ is the displacement and $\omega_E$ is the frequency of the oscillation.

Einstein argues that the energy levels of each one of the oscillators are given according to quantum mechanics, are discrete, and are: $\epsilon_r = \hbar \omega_E (r + 1/2) $, with r=0,1,2...

Thus the partition function is $z_1 =\sum_r e^{-\beta \epsilon_r}= {e^{-\beta \hbar \omega_E \over 2} \over 1- e^{-\beta \hbar \omega_E} }$, where $\beta = 1/kT $

From here we find the average energy and the thermal capacity.

Question: The above analysis seems to treat the oscillators as distinguishable. But aren't they indistinguishable? I have read two thinks:

1)That the Gibbs paradox can be resolved by dividing with N!. Thus, if the oscillators are indistinguishable, shouldn' we divide the the partition function with (3N)! ?

2) I have also read(from post here and papers) that the division of the partition function is not doe to the fact that quantum mechanically the particles should be identical, but because of our definition of entropy in thermodynamics(and that the Gibbs paradox can be resolved so).

Thus, I would like to ask, why don't we divide, in the Einstein model with the factorial, and why,without the division, the Einstein model works giving good results.

Thank you.

Note: The book I read from is of Mandle. Also, if there models like the Einstein model which consider the oscillators identical, if you can give a reference. Of course any reference for study is welcomed.

Constantine Black
  • 1,604

1 Answers1

4

Question: The above analysis seems to treat the oscillators as distinguishable. But aren't they indistinguishable?

In Einstein's model, the oscillators are supposed to sit at (oscillate around) definite place in space. So you could say they are distinguishable. For example, by their cartesian coordinates with respect to lab frame.

1)That the Gibbs paradox can be resolved by dividing with N!.

Unfortunately, the term "Gibbs paradox" is very vague and people use this term in different meanings. What do you mean by it?

I would like to ask, why don't we divide, in the Einstein model with the factorial,

The Einstein model treats the oscillators as independent systems and describes them by canonical ensemble - or, effectively, with the Boltzmann distribution of probability. In this context, the partition function for a system of $N$ oscillator is defined by $$ Z= \sum_{R} e^{-\beta \epsilon_R} $$ where the sum is over all states $R$ of the whole system, so that the Boltzmann probability can be expressed simply as $$ p_R = \frac{e^{-\beta \epsilon_R}}{Z}. $$

There is no reason to define $Z$ with above sum divided by $N!$ - if we did that, we would have to write the probability for state $R$ as $$ p_R = \frac{e^{-\beta \epsilon_R}}{N!Z} $$ which is cumbersome and serves no useful purpose.

and why,without the division, the Einstein model works giving good results.

Einstein was a clever guy and got lucky inventing a calculation that gives results similar to measurements.

Ján Lalinský
  • 44,745