Landau: Explaining the Definition of "Number of States with Energy

[Stalafin]

Question about Landau: Definition of "Number of states with energy" in an interval

Hey! I am currently reading Landau's Statistical Physics Part 1, and in Paragraph 7 ("Entropy") I am struggling with a definition.

Right before Equation (7.1) he gives the "required number of states with energy between $E$ and $E+\mathrm{d}E$" as:
$$\frac{\mathrm{d}\Gamma(E)}{\mathrm{d}E} \mathrm{d}E$$

I don't understand this equation. Am I supposed to understand $\Gamma(E)$ as a continuous function, and therefore $\frac{\mathrm{d}\Gamma(E)}{\mathrm{d}E}$ as a derivative?

Furthermore, how is the energy probability distribution
$$W(E) = \frac{\mathrm{d}\Gamma(E)}{\mathrm{d}E} w(E)$$
different from $w(E)$. Isn't $w(E)$ kind of a probability distribution by itself?

 

[marcusl]

For your first question, yes. For the enormous ensembles considered by stat mech (typically 10^23 particles or larger), the number of discrete states approaches infinity so their distribution may be considered continuous.

For the second question, what is w(E)? I don't have access to the text right now...

 

[Stalafin]

Landau considers that part for a quantum mechanical system. $w_n = w(E_n)$ is the distribution function for the system.

He gets there from the diagonal elements of the density matrix $w_n = w_{nn}$ (since the statistical distributions must be stationary), which can be expressed as functions of the energy levels alone (assuming we have a system in a coordinate system, such that it is at rest and apart from the energy the other integrals of motion don't factor in) $w_n=w_(E_n)$.

 

[marcusl]

I think this is the usual distinction discussed in statistics between probability P and probability density function (PDF) p. Here W(E)dE is the probability of seeing the system in a state with energy between E and E+dE. It is given in terms of the product of the PDF w(E) and the number of states $d\Gamma(E)$ in that interval.