Editing Monte Carlo in the microcanonical ensemble
Jump to navigation
Jump to search
The edit can be undone. Please check the comparison below to verify that this is what you want to do, and then publish the changes below to finish undoing the edit.
Latest revision | Your text | ||
Line 3: | Line 3: | ||
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: | where: | ||
Line 12: | Line 12: | ||
The first term on the right hand side | |||
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). | the [[Potential energy | potential energy]] (a function of the positional coordinates). | ||
Line 24: | Line 29: | ||
\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: |