Monte Carlo in the microcanonical ensemble: Difference between revisions
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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: | ||
where the first term on the right hand side is the [[Kinetic energy |kinetic energy]], whereas the second | : <math> H(X^{3N},P^{3N}) = \sum_{i=1}^{3N} \frac{p_i^2}{2m} + U \left( X^{3N} \right) </math>; (Eq.1) | ||
the [[Potential energy | potential energy]] (a function of the positional coordinates) | |||
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 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]]; | 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 | ||
| Line 17: | Line 24: | ||
\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. 2) | ||
where <math> \left. | 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. | 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> ; | <math> r = \left. \sqrt{ 2 m \Delta E } \right. </math> ; | ||
therefore: | therefore: | ||
Latest revision as of 11:20, 4 July 2007
Integration of the kinetic degrees of freedom[edit]
Consider a system of identical particles, with total energy given by:
- ; (Eq.1)
where:
- represents the 3N Cartesian position coordinates of the particles
- stands for the the 3N momenta.
The first term on the right hand side of (Eq. 1) is the kinetic energy, whereas the second term is
the potential energy (a function of the positional coordinates).
Now, let us consider the system in a microcanonical ensemble; let be the total energy of the system (constrained in this ensemble).
The probability, of a given position configuration , with potential energy can be written as:
- ; (Eq. 2)
![{\displaystyle \Pi \left(X^{3N}|E\right)\propto \int dP^{3N}\delta \left[K(P^{3N})-\Delta E\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a4f2b8463932fb289d4087b0d19a79e9a1c97bd)
where:
- is the Dirac's delta function
- .
The Integral in the right hand side of (Eq. 2) corresponds to the surface of a 3N-dimensional () hyper-sphere of radius ; therefore:
- .
![{\displaystyle \Pi \left(X^{3N}|E\right)\propto \left[E-U(X^{3N})\right]^{(3N-1)/2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d1348dde456f0c5c5b3411e85a773590b000dd3)
See Ref. 1 for an application of Monte Carlo simulation using this ensemble.