Monte Carlo in the microcanonical ensemble

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:

$H=\sum _{i=1}^{3N}{\frac {p_{i}^{2}}{2m}}+U\left(X^{3N}\right).$

where the first term on the right hand side is the kinetic energy, whereas the second one is the potential energy (a function of the positional 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:

$\Pi \left(X^{3N}|E\right)\propto \int dP^{3N}\delta \left[K(P^{3N})-\Delta E\right]$ ; (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 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 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.

References

1. N. G. Almarza and E. Enciso "Critical behavior of ionic solids" Phys. Rev. E 64, 042501 (2001) [4 pages ]