Monte Carlo in the microcanonical ensemble

Integration of the kinetic degrees of freedom

Consider 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)

Now, let us consider the system in a microcanonical ensemble; Let $\left.E\right.$ be the total energy of the system (conestrained in this ensemble)

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 $r=\left.{\sqrt {2m\Delta E}}\right.$ ; Therefore:

$\Pi \left(X^{3N}|E\right)\propto \left[E-U(X^{3N})\right]^{(3N-1)/2}$

See Ref 1 for an application of Monte Carlo simulation using this ensemble.

References

1. N. G. Almarza and E. Enciso "Critical behavior of ionic solids" Phys. Rev. E 64, 042501 (2001) [4 pages ]