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

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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 [itex]E[/itex] and [itex]E+\mathrm{d}E[/itex]" as:
[tex]\frac{\mathrm{d}\Gamma(E)}{\mathrm{d}E} \mathrm{d}E[/tex]

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

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

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marcusl
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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...
 
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Landau considers that part for a quantum mechanical system. [itex]w_n = w(E_n)[/itex] is the distribution function for the system.

He gets there from the diagonal elements of the density matrix [itex]w_n = w_{nn}[/itex] (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) [itex]w_n=w_(E_n)[/itex].
 
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marcusl
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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 [itex]d\Gamma(E)[/itex] in that interval.
 

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