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 == | ||
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> | |||
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) | |||
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 | |||
The probability, <math> \left. \Pi \right. </math> of a given position | |||
<math> U \left( X^{3N} \right) </math> can be written as: | <math> U \left( X^{3N} \right) </math> can be written as: | ||
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\int d P^{3N} \delta \left[ K(P^{3N}) | \int d P^{3N} \delta \left[ K(P^{3N}) | ||
- \Delta E \right] | - \Delta E \right] | ||
</math> ; (Eq. | </math> ; (Eq. 1) | ||
where | 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 | 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. \sqrt{ 2 m \Delta E } \right. </math> ; | <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> \Pi \left( X^{3N}|E \right) \propto \left[ E- U(X^{3N}) \right]^{(3N-1)/2} | ||
</math>. | </math> | ||
See Ref 1 for an example of this method | |||
== References == | |||
#[http://dx.doi.org/10.1103/PhysRevE.64.042501 N. G. Almarza and E. Enciso "Critical behavior of ionic solids" Phys. Rev. E 64, 042501 (2001) [4 pages] ] | |||
PEOPLE AT WORK, SORRY FOR ANY INCONVENIENCE |