Editing Monte Carlo in the microcanonical ensemble

== Integration of the kinetic degrees of freedom ==

Consider a system of <math> \left. N \right. </math> identical particles, with total energy <math> \left. H \right. </math> given by:

: <math> H = \sum_{i=1}^{3N} \frac{p_i^2}{2m} + U \left( X^{3N} \right), </math>


where:

* <math>  \left. X^{3N} \right. </math> represents the 3N Cartesian position coordinates of the particles
* <math>  \left. P^{3N} \right. </math> stands for the  the 3N momenta.







The first term on the right hand side is the [[Kinetic energy |kinetic energy]], whereas the second one is
the [[Potential energy | potential energy]] (a function of the positional coordinates).

Now, let us consider the system in a [[Microcanonical ensemble |microcanonical ensemble]]; 
let <math> \left. E  \right. </math> be the total energy of the system (constrained in this ensemble).

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

: <math> \Pi \left( X^{3N}|E \right) \propto 
\int d P^{3N} \delta \left[ K(P^{3N}) 
- \Delta E \right]
</math> ;   (Eq. 1)
where <math> \left. P^{3N} \right. </math> stands for the <math>3N</math> momenta, and

: <math> \Delta E = E - U\left(X^{3N}\right) </math>.

The Integral in the right hand side of Eq. 1 corresponds to the surface of a 3N-dimensional hyper-sphere of radius 
<math> r = \left. \sqrt{ 2 m \Delta E } \right.  </math> ;
therefore:

:<math> \Pi \left( X^{3N}|E \right) \propto \left[ E- U(X^{3N}) \right]^{(3N-1)/2}
</math>.

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

[[Category: Monte Carlo]]

== References == 


#[http://dx.doi.org/10.1103/PhysRevE.64.042501 N. G. Almarza and E. Enciso "Critical behavior of ionic solids"  Physical  Review E 64, 042501 (2001) (4 pages) ]