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)