Monte Carlo in the microcanonical ensemble

Integration of the kinetic degrees of freedom

Considering a system of $N$ identical particles, with total energy $H$ given by:

$H=\sum _{i=1}^{3N}\frac{p_i^2}{2m}+U(X^{3N}).$

where the first term on the right hand side is the kinetic energy, whereas the second one is the potential energy (function of the position coordinates)

Let $E$ be the total energy of the system.

The probability, $\mathrm{\Pi }$ of a given position configuratiom $X^{3N}$, with potential energy $U(X^{3N})$ can be written as:

$\mathrm{\Pi }(X^{3N}|E)\propto \int d{P}^{3N}\delta [K(P^{3N})-\mathrm{\Delta }E]$ ; (Eq. 1)

where $P^{3N}$ stands for the 3N momenta, and

$\mathrm{\Delta }E=E-U(X^{3N})$

The Integral in the right hand side of Eq. 1 corresponds to the surface of a 3N-dimensional hyper-sphere of radious $r=2m\mathrm{\Delta }E$ ; Therefore:

$\mathrm{\Pi }(X^{3N}|E)\propto {[E-U(X^{3N})]}^{3N-1}$

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