Microcanonical Ensemble - probability of finding momentum

Question

I was trying to calculate the probability of finding a particle with momentum $p$ in the microcanonical ensemble in a 3-dimensional box.

$$\rho_i(\vec p)= \langle\delta(\vec p - \vec p_i)\rangle= \int \mathrm d^3q_1\ldots \mathrm d^3q_N\int \mathrm d^3 p_1\ldots \mathrm d^3p_N \;\rho(\vec q_1,\ldots,\vec p_1,\ldots,\vec p_i,\ldots,\vec p_N)\delta(\vec p-\vec p_i). $$

In the microcanonical ensemble, the phase space density is constant, so our work is to evaluate the last integral over the momentum space. Does the second integral correspond to a volume of a 3 dimensional "hypersphere" or a $3N$ dimensional hypersphere of radius $p$?

Answer 1

If hamiltonian of the system is (an harmonic oscillator)

\begin{equation} \mathcal{H} = \sum\limits_{i= 0}^{3N-1} \left( \frac{p_i^2}{2m} + 2m\pi^2 \nu^2q_i^2\right) \end{equation}

using the coordinate transformation

\begin{equation} x_i = \sqrt{2m}\pi \nu q_i \qquad \qquad x_{3N+i} = \frac{p_i}{\sqrt{2m}} \end{equation}

the phase space is converted to an hipersphere of 6N dimensions.

Answer 2

$$\rho_i(\vec p)= \langle\delta(\vec p - \vec p_i)\rangle= \int \mathrm d^3q_1\ldots \mathrm d^3q_N\int \mathrm d^3 p_1\ldots \mathrm d^3p_N \;\rho(\vec q_1,\ldots,\vec p_1,\ldots,\vec p_i,\ldots,\vec p_N)\delta(\vec p-\vec p_i). $$

The second integral is integrate over a $3N-3$ space for $r=\infty$.

Only the $i$th particle is pinned by the given vector $\vec p_i = \vec p$. All other space for $j=1,2,3, ..,i-1, i+1, ..,N$ will be integrated.

If you intend to integrate over a $3N$ surface of hyper-sphere of radius $p$, the $\delta$ function should be written as:

$$ \delta\left( p - \sqrt{\sum_j |\vec p_j|^2}\right). $$

This is typical form for microcanonical ensemble, bacause the total energy $$E = \sum_j \frac{|\vec p_j|^2}{2m} \,\,\,\Rightarrow \,\,\, p = \sqrt{2mE} =\sqrt{\sum_j|\vec p_j|^2}$$