The variation of the information content of a large Einstein solid

[Ted Ali]

Homework Statement

Calculate the information variation $\Delta I$ of a large Einstein Solid, when the number $N$ of quantum harmonic oscillators decreases and $q>>N$. Where $q$ is the total number of energy quanta and $q$ is fixed.

Relevant Equations

The internal energy $U$ is solely dependent on the number of energy quanta $q$. $U = qhf \text{ } (1)$. (The Schroeder approach).

For $q >> N$. $\Omega \approx \left( \frac{eq}{N} \right)^N \text{ } (2)$ (Schroeder, An introduction to thermal physics (2.21)).

Can we argue that: $\Delta I = - \Delta S \text{ } (3)?$
How large can $\Delta N$, be?

Thank you for your time.

 

[Ted Ali]

Attempt to a solution: $$dU = TdS - PdV + \mu dN \text{ } (4).$$
Since $q = constant$ we have from equation (1), that $dU = 0 \text{ } (5)$. Also $PdV = 0 \text{ } (6).$
As a result $$TdS = - \mu dN \text{ } (7).$$
But $\mu = - T \left( \frac{\partial S}{\partial N} \right) \text{ } = -kT\ln(1 + \frac{q}{N}) \text{ } (8).$
Since $q >> N:$ $$\mu = -kT\ln(\frac{q}{N}) \text{ } (9).$$
So, $(7)$ becomes $$dS = k\ln(\frac{q}{N})dN \text{ } (10).$$

Questions:
1. Can (and should) we integrate equation $(10)$ in order to get $\Delta S \text{ }?$
2. How large can $\Delta N$ be?
3. Can we say that $\Delta I = - \Delta S \text{ }?$ Where $I$ is the information content of our system.

Thank you for your time,
Ted.