This question is closely related to this one: Why is the density of states required conceptually? Should it be seen as a mathematical trick related to Fourier series?
But it was suggested that I ask this question separately.
My question is about the motivation of defining density of states, and its relationship with the boundary conditions when starting from a discrete situation.
I take a simple example: free propagating waves in 1D, thus a solution of the equation:
$$\partial_x^2 f - \frac{1}{c^2} \partial_t^2 f=0$$
I want to find the density of states and then compute the average energy at temperature $T$ that those waves are containing.
If I want to start from a discretized scenario before going to the continuum, I first have to choose boundary conditions on a "box size" $L$.
Periodic boundary choice:
I allow my waves to verify $f(x+L,t)=f(x,t)$. The wavevectors describing those waves are:
$$k_n=\frac{2 n \pi}{L}, n \in \mathbb{Z}$$
$$\langle E \rangle = 2\sum_{n \geq 0} \hbar \omega_n n_B(\omega_n)$$
Where $\omega_n = c |k_n|$, and $n_B(\omega_n)$ the boson population at frequency $\omega_n$ (and at temperature $T$). The factor "2" is here because of the forward and backward propagating waves. Also, the "gap" between two consecutives frequencies is $\delta \omega_n=2c \frac{\pi}{L}$. Using this remark I will be able to find the continuum limit of this summation. I have:
The factor $2$ is here because of the two directions of propagation. Now, if I want to "go to the continuum", I use the fact that: $E_n = \hbar \omega_n$, and:
$$\sum_{n \geq 0} f_n =\sum_{n \geq 0} f_n \frac{\delta E_n}{\delta E_n}=\frac{L}{2 \pi \hbar c}\sum_{n \geq 0} f_n \delta E_n$$
Thus, I have, for $L \gg 1$
$$ \langle E \rangle \approx \int_0^{+\infty} d E (2*\frac{L}{2 \hbar \pi c}) E \ n_B(E) = \frac{L}{\hbar \pi c} \int_0^{+\infty} d E \ E \ n_B(E)$$
It gives me the density of states: $\rho_L(E)=\frac{L}{ \hbar \pi c}$
My questions
- How can the density of state be physical as it depends on a fictive length $L$
If you agree with what I wrote, and if the density of states is something physical, I don't understand how it can be here. Indeed it explicitely depends on the fictive length $L$, it cannot be physical because of that. Now, as suggested in some of the answers, in the end of the calculation one has to take $L \to + \infty$. In this case the density of states would diverge. How to make sense of this quantity then ? Is it that we reason with the density of state "per" unit length (which does not diverge). Also, in this example it works, but are we sure that "in general", for more complicated scenarios, the density of state per unit volume would become independent of the box size that has been choosen ?
- How to avoid this "trick" of using periodic boundary condition in order to find $\langle E \rangle$
As suggested by some of the answers of the linked post, I could directly work in the continuum by considering delta distribution to define the density of states. It should in the end give the same result. I have no idea of how to proceed to find the same result directly working with the continuum and I am interested to see how the two approaches match. I guess that some "dirac distribution" in the limit $L \to +\infty$ in my calculation but I really don't see how.
- Does the choice of boundary conditions matter for the density of states ?
I tried to impose strict boundary conditions, i.e: $f(0,t)=f(L,t)=0$ which gives stationnary modes. The meaning of the modes differ but in the end of the calculation, I found the same density of states and average energy. Is it exactly what is meant by those topics Why are periodic boundary conditions used for the derivation of phonons? Why are periodic boundary conditions used for the derivation of phonons? The predictions I will make are independent of the boundary condition I will choose ? Is there a nice reference in which this thing is precisely discussed ? In all book that I found they just apply periodic boundary condition without any real justification for that.