Reduced units
The PumMa code is historically designed to be used with reduced units only. However, since the spring of 2007, PumMa now uses SI-units at the front end of the program. Internally, PumMa still uses reduced units to enhance computation.
There used to be several reasons to use reduced units instead of SI-units. The most prominent reason, historically, is that the use of reduced units allows us to use the law of corresponding states. This law states that the same set of units can be used to calculate many different states with different units. For instance with the same set of reduced units one simulation can be used to describe the behavior of argon, at a temperature of 60 K and a density of 840 kg/m3, and xenon, at a temperature of 112 K and a density of 1617 kg/m3. However, this law is only valid for systems consisting of only one particle type. For most current systems this is clearly not the case.
Currently, two other reasons to use reduced units internally are far more important. When using reduced units all numeric values within a simulation are at the order of magnitude of one. This makes it easier to spot errors in the simulation, since extreme values are very unlikely to occur. Moreover, using numbers in this order of magnitude ensures that no floating point errors occur, due to limited precision of the processor.
The choice of reduced units
The system of reduced units consists of four parameters:
- the unit for length σ*,
- the unit for mass m*,
- the unit for energy ε* and
- the unit for charge q*.
Out of these choices all other units follow. For example the unit of time is τ* = σ*√(m*/ε*). In the lipid model of Markvoort et al. the following set of reduced units has been used. These are also the default values for PumMa. In this table kB is Boltzmann's constant (kB = 1.38·10-23 J/K) and NA is Avogadro's constant (NA = 6.022·1023 mol-1).
| Quantity | Reduced unit | SI unit |
|---|---|---|
| Energy | ε* | 1.9665 kJ/mol = 0.47 kcal/mol |
| Length | σ* | 0.45 nm |
| Mass | m* | 56.11 amu = 93.173·10-27 kg |
| Charge | q* | 1 e = 1.602·10-19 C |
| Time | τ* = σ*√(m*/ε*) | 2.4037 ps |
| Temperature | T* = ε*/kB | 236.51 K |
| Pressure | P* = ε*/σ*3 | 358.35 bar = 3.5835·107 Pa |
| Mass density | ρ* = m*/σ*3 | 1022.5 kg/m3 |
| Number density | n* = 1/σ*3 | 10.974 nm-3 |
| Permittivity | ε0* = NA·q*2/(σ*ε*) | 1.7469·10-8 C2/Nm2 |
In the lipid model a temperature of 1.3 T* has been used, which is equivalent to 307 K. The system was also modeled at atmospheric pressure, using as the target pressure 0.00283 P*. Time steps of of 0.005 τ* were used in the integration, which equals a 12 fs real time step size.
Conversion Tool
Conversion from reduced units to SI units is not always as straightforward. Especially with the derived units. In order to make the conversion as simple as possible a conversion tool has been added to this page. On the left it is possible to choose the SI units of the four basic parameters. Using the Derive button gives you the SI units of the derived units, such as τ*, T*, P*, ρ*, n* and ε0*.
For instance, if one wants a time step of 1 fs, the corresponding step size in reduced units is given by dividing 1 fs by the value given in the top right box (2.404 ps for the default PumMa values). The result gives the the time step in reduced units (for the default PumMa values this equals 1/2404 = 0.00042 τ*).
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