Nondimensionalization and characteristic scales in physics

1 Introduction

When doing calculations in computational physics, it is usually necessary to convert equations to a nondimensional form, as a computer almost always works with dimensionless numbers. The most simple way to do this would probably be to equate the SI units (kilogram, meter, second…) to 1. This choice of scale, however, is not always appropriate as the interesting features of the dynamics of the system can take place at lengths and time intervals that are very small or large compared to these units. Plots and tables of the computational results are easier to read when the values in them are numbers that don’t differ from 1 by too many orders of magnitude. Examples of different choices of scales are the use of natural units (where we set c = 1 and ћ=1) in high-energy physics, and atomic units in quantum chemistry, where an appropriate choice of length unit is the Bohr and that of energy is Hartree.

2 Simple harmonic motion

As an example of a classical mechanical system with a characteristic timescale, consider a 1D Hookean oscillator with spring constant k and mass M. The equation of motion for the single position variable x(t) is

 M \frac{d^2 x}{dt^2} = -kx .           (1)

The obvious choice of typical timescale in this case is of course the period of oscillation

T = 2 \pi \sqrt{\frac{M}{k}} ,           (2)

and the corresponding nondimensional time variable would be \tilde{t}=t/T. This scale could also be found by simple dimensional analysis, forming a product of powers of the parameters M and k, and requiring that the product has dimensions of time:

[M]^{\alpha}[k]^{\beta} = kg^\alpha \times \frac{kg^\beta}{s^{2\beta}}=kg^{\alpha + \beta}s^{-2\beta} = s         (second) .         (3)

Solving this for alpha and beta gives us a characteristic time M^{1/2}k^{-1/2}, which is the period T divided by 2π and is of the same order of magnitude as T. If we want to integrate the equation of motion numerically with the finite-difference method, we would probably choose the discrete time step to be at least about 10-15 times smaller than T, and when plotting the trajectory x(t) we would usually choose a range of t-coordinate that is only a few periods long, no matter what the initial position x(0) and velocity x’(0) of the oscillating body are.

On the other hand, the motion of the oscillator described above does not have a general characteristic length scale. If we have two trajectories where in the first one the total energy (kinetic + potential) of the oscillator is 1 Joule and in the second one it’s 100 Joules, it is not appropriate to use the same range of x-coordinate when plotting the functions x(t), because in the latter case the amplitude of the trajectory is 10 times larger than in the former. This fact can also be seen from a simple dimensional analysis, by noting that there is no product of powers of M and k that has dimension of length:

[M]^{\alpha}[k]^{\beta} \neq m  (meter), for any \alpha, \beta .

3 Linear motion with damping

As another simple mechanical system that has a characteristic timescale, consider linear 1D motion with damping (air resistance or friction). The equation of motion is

M\frac{d^2 x}{dt^2}=-b\frac{dx}{dt},              (4)

where the damping parameter b has dimensions of mass/time. Immediately we can see that the variable M/b has dimensions of time, but we don’t know its physical significance yet. Solving the DE for x(t) and its time derivative, we obtain:

x(t) = x_0 + \frac{Mv_0}{b} - \frac{Mv_0}{b}e^{-b(t-t_0 )/M}        (5)

x'(t) = v_0 e^{-b(t-t_0 )/M} = v_0 2^{-\frac{b(t-t_0 )}{M \log 2}}=v_0 2^{-\frac{t-t_0 }{t_{1/2}}}  .      (6)

where x_0 and v_0 are the position and velocity at time t = t_0, and

t_{1/2}=\frac{M\log 2}{b}     (7)

is the half-life of the velocity.

Usually when some variable decays exponentially with some half-life, be it the number of radioactive atoms in a sample or the amount of a foreign substance (medication, environmental toxin) in human body, we can say that for our purposes the variable has practically become zero after about 6-10 half-lifes. Therefore the parameter t_{1/2}, or the related parameter M/b (which we obtained by dimensional analysis) can be seen as the characteristic timescale for the motion in this system. Again, there is no single characteristic length for the system, as the distance the object travels from the initial position x_0 in some number of half-lives is linearly proportional to the initial velocity v_0.

4  A particle inside rigid box

What about mechanical systems where the motion happens in some characteristic length scale? The simplest example of this kind of behavior is 1D motion of an object that has been confined in some interval [0, L] of the x-axis by impenetrable walls at points x = 0 and x = L. This can be described by saying that the potential energy of the object is zero when 0 < x < L and it is infinite when x < 0 or x > L. In quantum mechanics this model system is called ”particle in a box”, but here we consider only the classical mechanical equivalent.

If the collisions of the object with the walls are elastic, the trajectory x(t) is a sawtooth function like the one plotted below:


Figure 1. The periodic classical trajectory of a 1D particle in a rigid box.

Now the interval the object moves in is always [0,L] which doesn’t depend on its initial position and velocity (as long as the initial position is inside the box and the initial velocity is non-zero). Therefore the obvious choice of length scale for the system is L. However, this system does not have a characteristic time scale, as the period of the motion is 2L/v_0 which depends on the initial conditions.

5 A system with both a characteristic length and time

How, then, would one construct a system where there exist characteristic scales in both length and time? By dimensional analysis, we can expect that the set of parameters a,b,c,... in the equation of motion should be such that some product of their powers, a^{\alpha} b^{\beta} c^{\gamma} \dots , has dimensions of time and some other similar product has dimensions of length.

One way to make such a system is to form an equation of motion that contains both a Hookean returning force and a term that depends on both position and velocity:

M\frac{d^2 x}{dt^2} = -b x \left( \frac{dx}{dt} \right)^2 - kx        (8)

Now we can deduce that the parameter b needs to have dimensions [mass][length]^{-2}. A parameter formed from M, b and k that has dimensions of length is

L = \sqrt{\frac{M}{b}}            (9)

and a characteristic time is the familiar

T = \sqrt{\frac{M}{k}},            (10)

which already appeared in the case of simple harmonic motion. From these we can form dimensionless length and time variables

\tilde{x} = \frac{x}{L}, \tilde{t} = \frac{t}{T},            (11)

and written using these variables, the equation of motion becomes

\frac{d^2 \tilde{x}}{d\tilde{t}^2} = -A x \left( \frac{d\tilde{x}}{d\tilde{t}} \right)^2 - B\tilde{x} ,          (12)

where A and B are dimensionless numbers defined by:

A = \frac{bL^2}{M}, B = \frac{k T^2}{M}.

So how do the trajectories x(t) of this system look like? The equation of motion can’t be solved analytically, but we can do a numerical solution with Mathematica, Matlab or some other program capable of that kind of calculations. Setting the values of the parameters to A=0.5 and B=1, and using the initial position \tilde{x}(0) = 0 and a set of different initial velocities x'(0) = 0, 10, 100 or 10000, the following graphs are obtained:



Figure 2. Trajectories of a particle in a velocity-dependent force field for initial dimensionless velocities of 1, 10, 100 or 10000.

From the plots in Fig. 2., it is obvious that the amplitude and period of the trajectory stay in the same orders of magnitude for a very large range of initial velocities/energies of the moving object. A 10^4 – fold increase in the dimensionless velocity causes only an about 6-fold increase in amplitude and an about 4-fold increase in frequency. The amplitude seems to increase only logarithmically with increasing velocity, i.e. slower than its any fractional power.

Note, however, that if the initial kinetic energy of the particle at t = 0 is small enough, the velocity-dependent term in the equation of motion can be neglected, and the system behaves like a simple harmonic oscillator for which the amplitude is directly proportional to the energy. To see the more interesting properties of this system, as in the graphs above, the energy needs to be large enough. In physics there are many situations where a system behaves differently in different energy scales, one example being a collision of atomic nuclei which can be described with simple Coulombic repulsion at low collision energies but will involve the possibility of nuclear fusion at larger energies where the particles can overcome the repulsive electrostatic force. The classical mechanical system here is a very simple toy model of this kind of behavior.

6 Applications

Many problems of applied physics where characteristic lengths and times and dimensionless quantities appear are related to the physics of fluid flow. A familiar dimensionless number that describes fluid motion is the Reynolds number (Re), which is related to the probability of turbulence occurring in a fluid system. Others include the capillary number (denoted Ca) and Eötvös number (denoted Eo), which describe the significance of surface tension in relation to other forces.

When a way to calculate the relevant length scales for a particular system are known, it is easier to choose a sufficiently fine discretization and a sufficiently large computational domain (intervals of the x- and t-axes) when integrating equations of motion numerically.

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