Monte Carlo in the microcanonical ensemble
Integration of the kinetic degrees of freedom
Consider a system of identical particles, with total energy given by:
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 be the total energy of the system (constrained in this ensemble)
The probability, of a given position configuration , with potential energy can be written as:
- ; (Eq. 1)
where stands for the momenta, and
The Integral in the right hand side of Eq. 1 corresponds to the surface of a 3N-dimensional hyper-sphere of radius ; therefore:
- .
See Ref 1 for an application of Monte Carlo simulation using this ensemble.