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| == Integration of the kinetic degrees of freedom == | | == Integration of the kinetic degrees of freedom == |
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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:
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| : <math> H(X^{3N},P^{3N}) = \sum_{i=1}^{3N} \frac{p_i^2}{2m} + U \left( X^{3N} \right) </math>; (Eq.1)
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| where:
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| * <math> \left. X^{3N} \right. </math> represents the 3N Cartesian position coordinates of the particles
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| * <math> \left. P^{3N} \right. </math> stands for the the 3N momenta.
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| The first term on the right hand side of (Eq. 1) is the [[Kinetic energy |kinetic energy]], whereas the second term is
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| the [[Potential energy | potential energy]] (a function of the positional coordinates).
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| Now, let us consider the system in a [[Microcanonical ensemble |microcanonical ensemble]];
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| let <math> \left. E \right. </math> be the total energy of the system (constrained in this ensemble).
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| 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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| <math> U \left( X^{3N} \right) </math> can be written as:
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| : <math> \Pi \left( X^{3N}|E \right) \propto
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| \int d P^{3N} \delta \left[ K(P^{3N})
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| - \Delta E \right]
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| </math> ; (Eq. 2)
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| where:
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| * <math> \left. \delta(x) \right. </math> is the [[Dirac delta distribution|Dirac's delta function]]
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| * <math> \Delta E = E - U\left(X^{3N}\right) </math>.
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| 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
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| <math> r = \left. \sqrt{ 2 m \Delta E } \right. </math> ;
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| therefore:
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| :<math> \Pi \left( X^{3N}|E \right) \propto \left[ E- U(X^{3N}) \right]^{(3N-1)/2}
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| </math>.
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| See Ref. 1 for an application of Monte Carlo simulation using this ensemble.
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| [[Category: Monte Carlo]]
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| == References ==
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| #[http://dx.doi.org/10.1103/PhysRevE.64.042501 N. G. Almarza and E. Enciso "Critical behavior of ionic solids" Physical Review E 64, 042501 (2001) (4 pages) ]
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