# The Hamiltonian

The Hamiltonian, ${\mathcal {H}}$, is a function that calculates the sum of the potential and kinetic energy of a system. For the systems considered here, the Hamiltonian is a function of the positions (r) and momenta (p) of the particles in the system. The potential energy, ${\mathcal {V}}$, depends only on the positions of the atoms $({\textbf {r}})$ and the kinetic energy, ${\mathcal {T}}$, depends only on the momentum of the particles $({\textbf {p}})$ [1]. Therefore, the Hamiltonian can be separated into a potential energy term that depends only on the positions of the particles (${\mathcal {V}}({\textbf {r}})$) and a kinetic energy term that depends only on the momenta of the particles (${\mathcal {T}}({\textbf {p}})$),
${\mathcal {H}}({\textbf {r}},{\textbf {p}})={\mathcal {V}}({\textbf {r}})+{\mathcal {T}}({\textbf {p}})$
The microcanonical ensemble is the statistical ensemble used to represent the isolated system, where the number of particles (N), the volume (V), and the total energy (E) are constant.

# Phase Space

The unique positions and momenta of each particle, for a system of N particles, can be accurately described by a 6 N dimensional spaced called phase space. Each particle in a system has three unique position variables $(x,y,z)$ ranging from 0 to L and three unique momentum variables $(p_{x},p_{y},p_{z})$ ranging from $-\infty$ to $\infty$ [1]. These 6N numbers constitute the microscopic state of the system at time t [2,3]. A system of N particles corresponds to a phase space with 6 N variables, which describes every conceivable position and momentum combination of all particles in a mechanical system [1,2,3].

### Phase Point

A phase point refers to any one point in an N-body system at any time t [2]. The dynamics of the phase point are fully described by the motion and trajectory of the phase point as it travels through phase space [2]. Every classical state can be considered to be equally probable as N, V, and E are held constant [3].

### Example

Consider a system containing two particles of Argon (labelled 1 and 2, respectively), where N = 2. Since phase space is described by 6 N dimensions the total number of spatial variables becomes 6(2) = 12.
* r$_1$, position of particle 1: $x_{1},y_{1},z_{1}$
* r$_2$, position of particle 2: $x_{2},y_{2},z_{2}$
* p$_1$, momentum of particle 1: $p_{x1},p_{y1},p_{z1}$
* p$_2$, momentum of particle 2: $p_{x2},p_{y2},p_{z2}$
Argon is a noble gas and inert to intermolecular interactions with other particles in a system. Accordingly, the two Argon atoms in this hypothetical system are non-interacting. When atoms are assumed to be non-interacting, potential energy interactions between particles are no longer accounted for and ${\mathcal {V}}$ = 0.
The Hamiltonian becomes the sum of the kinetic energies of the two particles:
    ${\mathcal {H}}={\frac {p_{1}^{2}}{2m}}+{\frac {p_{2}^{2}}{2m}}$

# The Classical Partition Function

In quantum mechanical systems, the system must occupy a set of discrete (quantized) states. These states correspond to the energy levels of the system, labelled $E_{1},E_{2},...,E_{n}$ A partition function can be defined a sum over these states.
    $Q=\sum _{i}\exp \left({\frac {-E_{i}}{k_{B}T}}\right)$
Where $E_{i}$ is the energy of state $i$, $k_{B}$ is the Boltzmann constant and $T$ is the temperature.
Explicit sums are practical when the energy states of molecules are independent and there are no intermolecular interactions. In systems where intermolecular interactions are significant, it is generally more practical to approximate the system as existing in terms of continuous variables. The states are approximated to be continuous and can be integrated with respect to the phase space of the system while simultaneously sampling every state of the system. This leads to the classical partition function:
    $Q_{classical}=\int \dots \iint \dots \int \dots \iint \exp \left({\frac {-{\mathcal {H}}({\textbf {r}},{\textbf {p}})}{k_{B}T}}\right)\,dr_{1}\,dr_{2}\dots \,dr_{N}\,dp_{1}\,dp_{2}\dots \,dp_{N}$

### Many Body Systems

The expectation values of a classical system can be evaluated by integrating over phase space. The classical partition function is an integral over 6N phase space variables:
    ${\textrm {d}}{\textbf {r}}=dx_{1}\,dy_{1}\,dz_{1},dx_{2},dy_{2},dz_{2},...,dx_{N}\,dy_{N}\,dz_{N}$
    ${\textrm {d}}{\textbf {p}}=dp_{x,1}\,dp_{y,1},dz_{z,1},...,dp_{x,N}\,dp_{y,N}\,dp_{z,N}$$Q_{classical}=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{0}^{L}\int _{0}^{L}\int _{0}^{L}\exp \left({\frac {-{\mathcal {H}}({\textbf {r}},{\textbf {p}})}{k_{B}T}}\right){\textrm {d}}{\textbf {r}}{\textrm {d}}{\textbf {p}}$

## References

* McQuarrie, Donald A. 1973. Statistical Thermodynamics. New York, N.Y. Harper & Row, Publishers, Inc. Pages 117-118.
* Tuckerman, Mark E. 2010. Statistical Mechanics: Theory and Molecular Simulation. Oxford England. New York: Oxford University Press. Pages 2-5.