softmatter:Simulation Variables/Units

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Reduced units

Dimensionless variables, or reduced units, are typically utilized in simulation. That is, quantities such as temperature, density, pressure, etc. are expressed in terms of convenient units of energy, length, and mass. One natural and convenient (but not unique) choice of units is

  • Unit of energy, ε
  • Unit of length, σ
  • Unit of mass of a particle, M

Other relevant units can be obtained from combinations of these units. For example,

  • Unit of time, τ = σ(M / ε)1 / 2
  • Unit of temperature, [T] = ε / kBT

Here kB is Boltzmann's constant. All simulation variables can then be written in reduced units by dividing by the appropriate basic unit. Examples are given in the table below.

Dimensionless variables
symbol meaning definition
r* dimensionless distance r / σ
E* dimensionless energy E / ε
T* dimensionless temperature kT / ε
ρ * dimensionless number density ρσ3
U* dimensionless internal energy U / ε
t* dimensionless time t / [σ(M / ε)0.5]
v* dimensionless velocity v / (ε / M)0.5
F* dimensionless force Fσ / ε
P* dimensionless pressure Pσ3 / ε
D* dimensionless self diffusion coefficient D / [σ(ε / M)0.5]


The reason behind using reduced units is that infinitely many combinations of density, temperature, particle diameter, mass, energy, etc. all correspond to the same state in reduced units. This is known as the law of corresponding states. For example, a simulation using the Lennard-Jones Potential at reduced units of ρ * = 0.5 and T * = 0.5 corresponds to both Argon at a state point characterized by T = 60K, ρ = 840kg / m3and Xenon at a state point characterized by T=112K and ρ = 1617kg / m3.[1] Without the use of reduced units, certain universal features of fluids, such as the equivalence of these different systems at these two state points, would be missed.

There is another, practical, reason for using reduced units[1]. In an actual simulation run with SI units, the absolute numerical values of the computed quantites are typically very small or very large compared to one. When several such quantities are multiplied using standard floating point multiplication, there is a risk that, at some stage of the calculation, a result will create an overflow or underflow. In reduced units, however, nearly all quantities of interest are of low order, typically between, say, 0.001 and 1000. Thus if, during a simulation, a very large or very small number suddenly results (e.g. of the order of Avogadro's number or larger), then that should trigger the suspicion of an error in the calculation. With non-reduced units, this same error might be more difficult to detect.

Of course, simulation results obtained in reduced units can always be converted back to real (e.g. SI) units, using the above table and values for the basic units (m, ε, etc.) for the system of interest. Examples of basic units for various substances are given in the table below {2}.

LJ basic units
substance ε / kB(K) σ(Angstroms)
Acetone 560.2 4.600
Acetylene 231.8 4.033
Air 78.6 3.711
Ammonia 558.3 2.900
Argon 119.8 3.405 *these values from page 42, Ref. [1]
Benzene 412.3 5.349
Bromine 507.9 4.296
n-butane 310 5.339
i-butane 313 5.341
Carbon dioxide 195.2 3.941
Carbon disulfide 467 4.483
Carbon monoxide 91.7 3.690
Carbon tetrachloride 322.7 5.947
Carbonyl sulfide 336 4.130
Chlorine 316 4.217
Chloform 340.2 5.389
Cyanogen 348.6 4.361
Cyclohexane 297.1 6.182
Cyclopropane 248.9 4.807
Ethane 215.7 4.443
Ethanol 362.6 4.530
Ethylene 224.7 4.163
Fluorine 112.6 3.357
Helium 10.22 2.551
n-hexane 339.3 5.949
Hydogen 59.7 2.827
Hydrogen cyanide 569.1 3.630
Hydrogen chloride 344.7 3.339
Hydrogen iodide 288.7 4.211
Hydrogen sulfide 301.1 3.623
Iodine 474.2 5.160
Krypton 178.9 3.655
Methane 148.6 3.758
Methanol 481.8 3.626
Methylene chloride 356.3 4.898
Methyl choride 350 4.182
Mercury 750 2.969
Neon 32.8 2.820
Nitric oxide 116.7 3.492
Nitrogen 71.4 3.798
Nitrous oxide 232.4 3.828
Oxygen 106.7 3.467
n-Pentane 341.1 5.784
Propane 237.1 5.118
n-Propyl alcohol 576.7 4.549
Propylene 298.9 4.678
Sulfur dioxide 335.4 4.112
Water 809.1 2.641


For Argon, M = 0.03994 kg/mol (from the periodic table of elements). Looking at the table above, we see that ε / kB = 119.8 K and σ = 3.405 Angstrom . From these three basic units we can convert reduced units into SI units for Argon:

temperature: T* = 1 corresponds to T=119.8 K

density: ρ * = 1 corresponds to ρ = 0.02533 molecules/angstrom3 = 1680 kg / m3

time: Δt * = 0.005 corresponds to Δt = 10.9 fs

pressure: P* = 1 corresponds to P = 41.9 MPa

Important Note on Density

In simulation, density typically refers to number density, which is the number of particles per volume. When dealing with reduced units, one must convert the real density into molecules/volume by either:

  • dividing by the mass of a molecule when given mass/volume.
  • multiplying by Avogadro's number when given moles/volume.


[1] D. Frenkel and B. Smit, Understanding molecular simulation : from algorithms to applications, 2002.

[2] Table reproduced from J.O. Hirschfelder, C.F. Curtiss and R.B. Bird, Molecular Theory of Gases and Liquids, New York, Wiley, 1954.

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