Natural scales

In making a mathematical model of a physical  situation, we typically have a number of dimensioned parameters involved: a mass, a spring constant, the gravitational field, .... We then have to carry out an analysis that calculates something for the given situation.

What we want from this analysis is typically some other dimensioned parameter: a distance, a velocity, an energy,.... In general, we start with a questions and some equations (physical relations among parameters and variables). We expect to do some mathematical manipulations and come up with an equation that gives an answer for whatever it is we are trying to calculate.

Since the goal of any problem is to express an unknown quantity in terms of known ones, our  solution will express an unknown dimensioned variable or parameter in terms of a set of dimensioned parameters that are presumed known.

In some cases, we have a limited set of known dimensioned parameters. There may turn out to be only one way of putting together our known parameters that will produce something with the dimensionality of what we are looking for. In this case, we know exactly what our answer is going to turn out to look like!  It could be multiplied by a dimensionless constant in front.  Our analysis of dimensions can't tell us anything about a fact of 2 or 26 or 0.01.  (Interestingly enough, the dimensionless constant often turns out to be on the order of 1, though we sometimes get 2π or 1/2π, depending on how we choose to describe our known parameters.) This can give us a plausible estimate for the scale of our unknown up to an order of magnitude. A link to a specific example is given in the Follow-ons.

When we combine the dimensioned parameters of our problem to create a mass (M), length (L), time (T), charge (Q), or temperature ($\Theta$) we refer to this as a natural scale for the problem. That is, it is a scale that can be created from the problem parameters. These natural scales have a number of useful characteristics to help us think about our problem and how to solve it.

If we can create a natural mass, $m_N$, a natural length $l_N$, and a natural time $t_N$, then we can also create a natural velocity $v_N = l_N/t_N$,  a natural acceleration $a_N = l_N/t_N^2$, a natural energy, etc. Any real solution for these quantities can be expected to be proportional to our natural quantities times some dimensionless number.

Another way of looking at this is to observe that any dimensioned parameter is constructed as a combination of a number of distinct measurements. Thus a velocity may be constructed from a distance measurement and a time measurement. An energy may be constructed from a measurement of a mass, a distance, and two times. When we have a set of parameters, they typically are not simply basic measurements of mass, length, and time. Constructing the "natural scales" gives a set of measurements of fundamental (operationally defined) quantities, which, when combined appropriately, will yield the given parameters of the system. (I am grateful to Adam Franklin for this insight.)

Joe Redish 9/22/11