Monte Carlo in the microcanonical ensemble
Integration of the kinetic degrees of freedom
Consider a system of identical particles, with total energy given by:
- ; (Eq.1)
- represents the 3N Cartesian position coordinates of the particles
- stands for the the 3N momenta.
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. 2)
- is the Dirac's delta function
The Integral in the right hand side of (Eq. 2) 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.