Monte Carlo in the microcanonical ensemble

Integration of the kinetic degrees of freedom

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

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

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 $\left.E\right.$ be the total energy of the system.

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

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

where $\left.P^{3N}\right.$ stands for the 3N momenta, and

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

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

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

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