Editing Monte Carlo in the microcanonical ensemble

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== Integration of the kinetic degrees of freedom ==
== 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:
Considering 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>


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


where:
Let <math> \left. E  \right. </math> be the total energy of the system.
 
The probability, <math> \left. \Pi \right. </math>  of a given position configuratiom <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 3N 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 radious
<math> r = \left. 2 m \Delta E \right.  </math> ;
Therefore:
 
:<math> \Pi \left( X^{3N}|E \right) \propto \left[ E- U(X^{3N}) \right]^{3N-1}
</math>


* <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 of (Eq. 1) is the [[Kinetic energy |kinetic energy]], whereas the second term 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. 2)
where:
* <math> \left. \delta(x) \right. </math> is the [[Dirac delta distribution|Dirac's delta function]]


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


The Integral in the right hand side of (Eq. 2) corresponds to the surface of a 3N-dimensional (<math> p_i; i=1,2,3,\cdots 3N </math>) 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) ]
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