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:

• $\left.X^{3N}\right.$ represents the 3N Cartesian position coordinates of the particles
• $\left.P^{3N}\right.$ stands for the the 3N momenta.

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

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

The probability, $\left.\Pi \right.$ of a given position configuration $\left.X^{3N}\right.$, 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 radius $r=\left.{\sqrt {2m\Delta E}}\right.$ ; 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" Physical Review E 64, 042501 (2001) (4 pages)