# Explanation of definition of entropy

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

with

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

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?