Microcanonical ensemble

Microcanonical ensemble

Ensemble variables

(One component system, 3-dimensional system, ... ):

• $\left.N\right.$: number of particles

• $\left.V\right.$: is the volume

• $\left.E\right.$: is the internal energy (kinetic + potential)

Partition function

$Q_{NVE}={\frac {1}{h^{3N}N!}}\iint d(p)^{3N}d(q)^{3N}\delta (H(p,q)-E).$

where:

• $\left.h\right.$ is the Planck constant

• 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( q \right)^{3N} } represents the 3N Cartesian position coordinates.

• 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 \right)^{3N} } represents the 3N momenta.

• 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 \left(p,q\right) } represents the Hamiltonian, i.e. the total energy of the system as a function of coordinates and momenta.

• 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 \left( x \right) } is the Dirac delta distribution

Thermodynamics

$\left.S=k_{B}\log Q_{NVE}\right.$

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. S \right. } is the entropy.

• 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. k_B \right. } is the Boltzmann constant

References

Related reading

• D. Frenkel and B. Smit, "Understanding Molecular Simulation: From Algorithms to Applications", Academic Press
• Philipp Schierz, Johannes Zierenberg and Wolfhard Janke "Molecular Dynamics and Monte Carlo simulations in the microcanonical ensemble: Quantitative comparison and reweighting techniques", Journal of Chemical Physics 143 134114 (2015)