# 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