Editing Monte Carlo in the microcanonical ensemble
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Consider a system of <math> \left. N \right. </math> identical particles, with total energy <math> \left. H \right. </math> given by: | 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 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) | |||
where | |||
the [[Potential energy | potential energy]] (a function of the positional coordinates) | |||
Now, let us consider the system in a [[Microcanonical ensemble |microcanonical ensemble]]; | 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 | The probability, <math> \left. \Pi \right. </math> of a given position configuration <math> \left. X^{3N} \right. </math>, with potential energy | ||
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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 <math>3N</math> 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 radius | ||
<math> r = \left. \sqrt{ 2 m \Delta E } \right. </math> ; | <math> r = \left. \sqrt{ 2 m \Delta E } \right. </math> ; | ||
therefore: | therefore: | ||
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#[http://dx.doi.org/10.1103/PhysRevE.64.042501 N. G. Almarza and E. Enciso "Critical behavior of ionic solids" | #[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] ] |