Monte Carlo in the microcanonical ensemble: Difference between revisions
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the potential energy (function of the position coordinates) | the potential energy (function of the position coordinates) | ||
Now, let us consider the system in a 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 ( | 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 | The probability, <math> \left. \Pi \right. </math> of a given position configuration <math> \left. X^{3N} \right. </math>, with potential energy | ||
<math> U \left( X^{3N} \right) </math> can be written as: | <math> U \left( X^{3N} \right) </math> can be written as: | ||
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: <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 | 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: | ||
Revision as of 16:59, 28 February 2007
Integration of the kinetic degrees of freedom
Consider a system of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. N \right. } identical particles, with total energy Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. H \right. } given by:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H = \sum_{i=1}^{3N} \frac{p_i^2}{2m} + U \left( X^{3N} \right). }
where the first term on the right hand side is the kinetic energy, whereas the second one is the potential energy (function of the position coordinates)
Now, let us consider the system in a microcanonical ensemble; Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. E \right. } be the total energy of the system (constrained in this ensemble)
The probability, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. \Pi \right. } of a given position configuration Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. X^{3N} \right. } , with potential energy Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U \left( X^{3N} \right) } can be written as:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Pi \left( X^{3N}|E \right) \propto \int d P^{3N} \delta \left[ K(P^{3N}) - \Delta E \right] } ; (Eq. 1)
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. P^{3N} \right. } stands for the 3N momenta, and
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta E = E - U\left(X^{3N}\right) }
The Integral in the right hand side of Eq. 1 corresponds to the surface of a 3N-dimensional hyper-sphere of radius Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r = \left. \sqrt{ 2 m \Delta E } \right. } ; Therefore:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Pi \left( X^{3N}|E \right) \propto \left[ E- U(X^{3N}) \right]^{(3N-1)/2} }
See Ref 1 for an application of Monte Carlo simulation using this ensemble.