Microcanonical ensemble

Microcanonical ensemble

Ensemble variables

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

• $N$: number of particles

• $V$: is the volume

• $E$: 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:

• $h$ is the Planck constant

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

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

• $H(p,q)$ represents 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

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)