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.

Thermodynamics

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

where:

• $\left. S \right.$ is the entropy.