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: | ||
: <math> H = \sum_{i=1}^{3N} \frac{p_i^2}{2m} + U \left( X^{3N} \right) | : <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 | 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) | 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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where <math> \left. P^{3N} \right. </math> stands for the <math>3N</math> momenta, and | 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> | : <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 radius | The Integral in the right hand side of Eq. 1 corresponds to the surface of a 3N-dimensional hyper-sphere of radius | ||
Revision as of 18:19, 28 February 2007
Integration of the kinetic degrees of freedom
Consider a system of identical particles, with total energy given by:
where the first term on the right hand side is the kinetic energy, whereas the second one 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. 1)
![{\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 stands for the momenta, and
- .
The Integral in the right hand side of Eq. 1 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.