Statistical Mechanics: Microcanonical State Probability

[alejandrito29]

In a aislate system, the probability on a microcanonical state $$\Gamma$$ is

$$p(\Gamma ) = 1/K$$ , if E<H<E + ΔE, and 0 on otherwise

with $$K = \int_{\Gamma : E<H<E+ΔE} d \Gamma$$

a) Show that ΔE →0, then
$$p(\Gamma) = \frac{\delta (E-H)}{\int_{\Gamma : H=E} \delta(E-H)}$$

b) Show that if use the change of variable $$\Gamma \to (X,a)$$, with $$X$$ are 6N-1 coordinates abaut the surface H=E, and a is a perpendicular coordinate to this surface at the point X, then

$$\int_D \delta (E-H) d \Gamma = \int_{D_E} \frac{ d X}{ || \frac{dH}{d \Gamma}||}$$




The rules of this forums says that i says my tried, but, sincerely i don't idea abaut this problem, i think on Taylor for the question a), but i don't have result.

 

[member 553083]

For the question b), i think on use polar coordinates, because the surface H=E is a sphere, but i don't arrive to the result.