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

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

• $\left( p \right)^{3N}$ represents the 3N momenta.

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

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

Thermodynamics

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

where:

• $\left. S \right.$ is the entropy.

• $\left. k_B \right.$ 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)