# Density of states in microcanonical ensemble for discrete and continuum energy spectrum

I'm introducing myself to statistical mechanics using two books: Introduction to Statistical Physics by S. Salinas, and Statistical Physics of Particles, by Mehran Kardar. Both textbooks work on an example of a isolated system which is composed of two systems which are allowed to exchange heat with each other only. System $$1$$ and $$2$$ have energy $$E_1$$ and $$E_2$$ respectively such that the total energy $$E_1+ E_2$$ is fixed( microcanonical ensemble). The systems have phase space volumes $$\Omega_1(E_1)$$ and $$\Omega_2(E_2=E- E_1)$$, respectively. In salinas book, it is assumed that the energy $$E_1$$ can assume only discrete values, so the phase space volume of the composite system is given by (eq 4.5 of textbook)

$$\Omega(E) = \sum_{E_1=0}^E \Omega_1(E_1)\Omega_2(E- E_1) \tag1$$

However, Mehran Kardar considers that the energy is a non-countable parameter, such that the total phase space volume is (eq. 4.4 of the textbook)

$$\Omega(E) = \int_0 ^E dE_1\Omega_1(E_1)\Omega_2(E- E_1) \tag2$$

My problem: This seems to be incorrect to me, since Integration over the energy $$E_1$$ should imply that the product $$\Omega_1 \Omega_2$$ would be some kind of density of phase space volume, and not the phase space volume by itself. Also we have that the dimension of $$\Omega(E)$$ in $$(1)$$ is different from the $$\Omega(E)$$ from $$(2)$$. What point am I missing?

Edit: As far as I have read about this topic, some texts define $$\Omega(E)$$ as the accessible "area" in system's phase space associated with energy $$E$$, but $$\Omega(E)$$ in actually the density of states of the system, not an area. Interpret $$\Omega(E)$$ as a density of microstates is ok to me since eq. $$(2)$$ means that $$\int dE_1 \Omega_1 \Omega_2$$ is also a density. But i'm quite confused about the discrete energy spectrum in a classical description, since the Interpretation of $$\Omega(E)$$ as a density in eq. $$(1)$$ makes no longer sense to me.

• I think that in $(2)$, $\Omega(E)$ should be seen as a density rather than a volume, i.e. $\Omega_1,\Omega_2$ are then also densities Commented Jan 19, 2022 at 16:30
• You are correct that Eq (2) is dimensionally inconsistent (given that $\Omega$ is unitless and $E$ is an energy. You can correct this by adding in a "small" energy scale $\Delta E$ and writing the integral like: $\int\frac{dE_1}{\Delta E} \Omega_1(E_1)\Omega_2(E - E_1)$
– hft
Commented Jun 12, 2023 at 16:06

The correct way to define the microcanonical partition function in continuous energy $$E$$ is this:
• Define $$\Omega(E)$$ (dimensionless) to be the number of microstates with energy anywhere between $$0$$ and $$E$$. The number of microstates in a narrow energy band $$dE$$ is $$d\Omega(E)$$.
• Define the microcanonical partition function $$\omega(E)$$ as $$\omega(E) = \frac{d\Omega(E)}{dE}$$
Clearly, $$\omega(E)$$ is a density, number of microstates per unit energy. The number of microstates in energy band $$dE$$ is $$d\Omega = \omega dE \Rightarrow \int d\Omega = \int \omega dE$$ Then all integrals in $$E$$ involve the density $$\omega$$, not $$\Omega$$.
I recommend using $$\Omega$$ when dealing with discrete systems and $$\omega$$ when dealing with continuous. This notation is not universally accepted but avoids confusion and I have adopted it in my lectures.