Monte Carlo in the microcanonical ensemble

Integration of the kinetic degrees of freedom

Consider a system of $\left.N\right.$ identical particles, with total energy $\left.H\right.$ given by:

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

where:

• $\left.X^{3N}\right.$ represents the 3N Cartesian position coordinates of the particles
• $\left.P^{3N}\right.$ 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 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. 2)

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. \delta(x) \right. } is the Dirac's delta function

• 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. 2) corresponds to the surface of a 3N-dimensional (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 p_i; i=1,2,3,\cdots 3N } ) 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:

$\Pi \left(X^{3N}|E\right)\propto \left[E-U(X^{3N})\right]^{(3N-1)/2}$.

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" Physical Review E 64, 042501 (2001) (4 pages)