Units¶

PYGAMD stores and computes all values in reduced units. The quantities in real units can be converted into the ones in reduced units by defining a set of fundamental units by user himself.

Fundamental Units¶

The three fundamental units are:

distance - \(\mathcal{\sigma}\)
energy - \(\mathcal{\varepsilon}\)
mass - \(\mathcal{m}\)

Temperature units (thermal energy)¶

PYGAMD accepts all temperature inputs and provides all temperature output values in units of energy: \(k_{B} T\), where \(k_{B}\) is Boltzmann’s constant. In reduced units, one usually reports the value \(T^* = k_{B}T/\mathcal{\varepsilon}\).

Charge units¶

The charge used in PYGAMD is also reduced. The units of charge are: \((4 \pi \epsilon_0 \epsilon_r \mathcal{\sigma} \mathcal{\varepsilon})^{1/2}\), where \(\epsilon_0\) is vacuum permittivity and \(\epsilon_r\) is relative permittivity.

With \(f= 1/4\pi \epsilon_0=138.935\text{ }kJ\text{ }mol^{-1}\text{ }nm\text{ }e^{-2}\), the units of charge are: \((\epsilon_r \mathcal{\sigma} \mathcal{\varepsilon}/f)^{1/2}\). Divide a given charge by this quantity to convert it into an input value for PYGAMD.

Common derived units¶

Here are some commonly used derived units:

time - \(\tau = \sqrt{\mathcal{m} \mathcal{\sigma}^2/\mathcal{\varepsilon}}\)
volume - \(\mathcal{\sigma}^3\)
velocity - \(\mathcal{\sigma}/\tau\)
momentum - \(\mathcal{m}\mathcal{\sigma}/\tau\)
acceleration - \(\mathcal{\sigma}/\tau^2\)
force - \(\mathcal{\varepsilon}/\mathcal{\sigma}\)
pressure - \(\mathcal{\varepsilon}/\mathcal{\sigma}^3\)

Example physical units¶

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

distance - \(\mathcal{\sigma} = \mathrm{nm}\)
energy - \(\mathcal{\varepsilon} = \mathrm{kJ/mol}\)
mass - \(\mathcal{m} = \mathrm{amu}\)

Derived units / values in this system:

time - picoseconds
velocity - nm/picosecond
pressure - 16.3882449645417 atm
force - 1.66053892103218 pN
\(k_{B}\) = 0.00831445986144858 kJ/mol/Kelvin