Microcanonical ensemble

Ensemble variables

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

• $\left.N\right.$: Number of Particles

• $\left.V\right.$: Volume

• $\left.E\right.$: 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:

• Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \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.

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

• Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \delta \left(x\right)} is the Dirac delta distribution

Thermodynamics

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.S=k_{B}\log Q_{NVE}\right.}

where:

• $\left.S\right.$ is the entropy

• $\left.k_{B}\right.$ is the Boltzmann constant

References

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