I am trying to understand the definition of the entropy. The lecture notes I use define it as $$ S(E,V) = k_B \ln(\Gamma(E,V)) $$


$$\Gamma(E) = \int_{E < H(p,q) < E + \Delta} d^{3N}p\ d^{3N}q\ \ \rho (p,q) $$

where $\rho$ is the distribution of $p$ and $q$ in the 6N-dimensional $\Gamma$ space of $p,q$ of a system with N elements. Now I have 3 questions:

  1. The $V$ in the definition, is it the volume, which corresponds to N?
  2. What is the $\Delta$ ?
  3. What do the integral boundaries mean? We look at all the Hamilton Functions with the Energry between $E$ and $E+\Delta$? But shouldn't the Hamilton Function for a certain system be unique? So why look at different energies? And what does it mean to integrate over Hamilton functions like that?

Okay, let me try to answer your questions one by one:

  1. This is correct. The V corresponds to the volume which in which the N particles are observed.

  2. The Delta means a finite element of energy. As mentioned below: it is an arbitrary, finite energy, which you later want to decrease until it is indefinite.

  3. Here you should keep in mind, what q,p,Г and H really mean. Г is the volume of the phase space which is occupied by the system with N particles which has the energy H(p,q). q & q are here 3N-dimensional coordinates of all relevant particles. If you compare this case with a normal 2D phase space you will easier see why you need to integrate like this: by observing a simple 1D-pendulum and its 2D phase space (which is a circle) you can get a good example. By allowing more energies and not only one, you will get an annulus.

  4. In case you're wondering why you integrate system between this boundaries: You're just doing this to subtract this integral over less-energy-states to get a finite differential of the phase space volume. By making Delta indefinite, you can write it as an derivative.

I hope this helps you to get a better understanding of this definition.

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