Microcanonical ensemble

Ensemble variables

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

• $N$: Number of Particles

• $V$: Volume

• $E$: Internal energy (kinetic + potential)

Partition function

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

where:

• $h$ is the Planck constant

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

• ${(p)}^{3n}$ represents the 3N momenta.

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

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

Thermodynamics

$S=k_B\log Q_{NVE}$

where:

• $S$ is the entropy.

• $k_B$ is the Boltzmann constant

References

1. D. Frenkel and B. Smit, "Understanding Molecular Simulation: From Algorithms to Applications", Academic Press