# Units¶

HOOMD-blue stores and computes all values in a system of generic, fully self-consistent set of units. No conversion factors need to be applied to values at every step. For example, a value with units of force comes from dividing energy by distance.

## Fundamental Units¶

The three fundamental units are:

- distance - \(\mathcal{D}\)
- energy - \(\mathcal{E}\)
- mass - \(\mathcal{M}\)

All other units that appear in HOOMD-blue are derived from these. Values can be converted into any other system of units by assigning the desired units to \(\mathcal{D}\), \(\mathcal{E}\), and \(\mathcal{M}\) and then multiplying by the appropriate conversion factors.

The standard *Lennard-Jones* symbols \(\sigma\) and \(\epsilon\) are intentionally not referred to here.
When you assign a value to \(\epsilon\) in hoomd, for example, you are assigning it in units of energy:
\(\epsilon = 5 \mathcal{E}\). \(\epsilon\) is **NOT** the unit of energy - it is a value with units of
energy.

## Temperature (thermal energy)¶

HOOMD-blue accepts all temperature inputs and provides all temperature output values in units of energy: \(k T\), where \(k\) is Boltzmann’s constant. When using physical units, the value \(k_\mathrm{B}\) is determined by the choices for distance, energy, and mass. In reduced units, one usually reports the value \(T^* = \frac{k T}{\mathcal{E}}\).

Most of the argument inputs in HOOMD take the argument name `kT`

to make it explicit. A few areas of the code
may still refer to this as `temperature`

.

## Charge¶

The unit of charge used in HOOMD-blue is also reduced, but is not represented using just the 3 fundamental units - the permittivity of free space \(\varepsilon_0\) is also present. The units of charge are: \((4 \pi \varepsilon_0 \mathcal{D} \mathcal{E})^{1/2}\). Divide a given charge by this quantity to convert it into an input value for HOOMD-blue.

## Common derived units¶

Here are some commonly used derived units:

- time - \(\tau = \sqrt{\frac{\mathcal{M} \mathcal{D}^2}{\mathcal{E}}}\)
- volume - \(\mathcal{D}^3\)
- velocity - \(\frac{\mathcal{D}}{\tau}\)
- momentum - \(\mathcal{M} \frac{\mathcal{D}}{\tau}\)
- acceleration - \(\frac{\mathcal{D}}{\tau^2}\)
- force - \(\frac{\mathcal{E}}{\mathcal{D}}\)
- pressure - \(\frac{\mathcal{E}}{\mathcal{D}^3}\)

## Example physical units¶

There are many possible choices of physical units that one can assign. One common choice is:

- distance - \(\mathcal{D} = \mathrm{nm}\)
- energy - \(\mathcal{E} = \mathrm{kJ/mol}\)
- mass - \(\mathcal{M} = \mathrm{amu}\)

Derived units / values in this system:

- time - picoseconds
- velocity - nm/picosecond
- k = 0.00831445986144858 kJ/mol/Kelvin