Units, Scaling, and functional dependence


Mathematical expressions in physics typically contain many symbols — lots of variables and parameters. How these variables and parameters come into an expression is a powerful tool for understanding how our mathematics connects to a physical system and what it says about it.

The way that an expression depends on a parameter is called functional dependence. How a mathematical expression changes (or the physical quantity is represents changes) when a parameter changes is called scaling. Understanding the dimensionality of an expression is the first critical step to figuring our how a system changes when the input parameters change.

In this page, we'll go through a set of simple examples to show how to work with dimensionality to see how scaling and functional dependence can give us  insights.

  • Units, dimensionality, and countingOur page on changing units shows how dimensionality helps us keep track our how to change units. Dimensionality starts by telling us not how a system changes, but how our equations change when we change our description (choice of unit). We'll give a simple example to show how different dimensionalities correspond to different functional dependences. 
  • Scaling: The dimensionality of a term in an equation or expression also tells us what happens to a system as something changes in a physical system. In biological systems, this can lead to an understanding of the implications and correlations between how systems change when parameters change. 
  • Comparing competing effects: The dimensionality of terms in an equation or expression tells us how important a parameter or variable is in producing a change.

Changing our choice of units: Counting, but counting what?

In seeing how an understanding of dimensionality leads to understanding functional dependence and scaling, let's go through how it works with changing units. 

In our discussion of dimensional analysis, we talked about what happens when we make a different choice of unit. In this case, the physical system doesn't change, but the number we assign to it does. For our equations to describe a relationship in a physical system that is true no matter how we choose to describe it, our dimensionalities (and units) have to match (have the same functional dependence). Let's see how this works in a simple example: length and area measurements.

An example: Length and area

Consider how we measure different kinds of physical space by comparing measurements of length and area. In both cases, we make a choice of a standard unit and count the number of times our unit fits in to get an assigned number.

In the case of length, we use a small line segment $\Delta x$, with dimensionality L:

$[\Delta x] =$ L.

The length only depends on a single power of a length measurement.

In the case of an area, it has a dimensionality L2 — as in a rectangle whose area equals the base times the height:

$[A] = [bh] =$ L2.

The area depends on two powers of a length measurement.

What this means is that instead of just using a line, we have to count how many times a small box fits into our area. Typically, we make the choice to make our area boxes squares with the side equal to our length segment. (This isn't absolutely necessary. Think about what we would have to do if we didn't do this.) 

One scale we chose when we were learning to estimate was to use a finger as length scale. One might choose a "thumb" unit (shown on the left below — about equal to 1 inch) or a "finger" unit (show on the right — about equal to 4 inches).

Let's call the thumb unit of length "1 th", and the finger unit of length "1 fi". Shown below is the result of what we get measuring a length using the two units. From the figure at the right, it's clear that the unit conversion rule is

                              4 th = 1 fi.

We measure a line by seeing how many of our units fit in. It's clear that we can express the length of the brown line below as either equal to 8 th or 2 fi.

This is clearly consistent with our conversion rule since

$$1 = \frac{4 \mathrm{th}}{1 \mathrm{fi}}$$

Multiplying 2 fi by 1 in this form gives 8 th. So the number of units just scales by the same factor that the unit changes by.

If we're looking at an area, we have to be a bit more careful. A line of 1 th length can fit into a box an infinite number of times. To count areas, we have to count squares of size (1 th) x (1 th) or (1 fi) by (1 fi). Let's see how this conversion works in a simple case.

To calculate an area, we have to see how many squares of unit size fit in. At the right are shown our basic units of area: (1 th) x (1 th) or (1 fi) by (1 fi). We write these respectively as 1 th2 or 1 fi2 respectively.

We can easily see how many of these fit into some area. At the left is shown an area that is clearly equal to 2 fi2 but is not 8, but 32 th2!

This makes sense if you just pay attention to what you are counting when you are making a measurement. If you are measuring a length, you are counting how many lengths of your unit fit in; if you are measuring an area, you are counting how many squares of your unit fit in, and if you are measuring a volume, you are counting how many cubes of your unit fit in. 

This is perhaps the simplest example we can give. It illustrates clearly why you have to scale an area by two factors of the scale conversion — and by extension why you have to scale a volume by three factors of the scale conversion.

In general, your dimensional analysis will tell you how many factors of your scale conversion you will need in any expression. If we change our length scale by a factor of $\lambda$, a quantity of dimensionality LN will change by a factor of $\lambda^N$. For practice, consider what factors you will need to change an area measured in nm2 (square nanometers) to m2 (square meters) and how this would differ from changing a volume measured in nm3 to one measured in m3.

Scaling: Resizing the system

Our functional dependences can play a more interesting role than just changing the way we choose to describe a physical system. They can help us understand how a physical system itself can change — how it scales with a change.

For example, the volume of an organism has dimensionality L3, while its surface area has dimensionality L2. This means that if an organism grows approximately isometrically (all dimensions changing by the same amount), then as it grows its volume will change differently than its area. If the organism increases each dimension by a factor of $\lambda$, then the volume will change by a factor of $\lambda^3$ and the area by a factor of $\lambda^2$.

An example: Scaling of basal metabolism

F. J. Ballesteros et al. Sci. Rept. 8:1448 (2018)

The different way that surface and volume change with size has serious implications for many biological situations. One group of researchers* argue that the rate of energy generation in an organism (basal metabolism) is a balance between the energy generated (proportional to the volume, dimensionality L3. and hence to the organism's mass, $M$) and the energy emitted through the organism's surface (dimensionality L2, and hence proportional to $M^{\frac{2}{3}}$). A model fit to the data, shown in the figure at the right, does quite well.

* F. J. Ballesteros et al., "On the thermodynamic origin of metabolic scaling," Scientific Reports 8, Article no 1448 (2018); doi.org/10.1038/s41598-018-19853-6.

For additional examples, see the problems Scaling up, and How big is a worm?

We'll also use scaling and functional dependence often to understand, for example, why emission of heat is a problem for large warm blooded animals but not for small ones, and to why in neutral ionic liquids, dipole forces play a dominant role. Darcy Thompson's 1917 book, On Growth and Form, provided an analysis for scaling of organisms with distortion that helped make sense of the role of development in evolution. 

Competing effects: Who wins?

Functional dependence is about what power of a particular measurement is involved in creating an algebraic expression. Those dependences play an powerful role in understanding the effect of parameter change on a system.

Often, a qualitative analysis of a system will tell us that there are two effects that happen: one that tends to increase the phenomenon we are looking at, the other to decrease it. Without the mathematics, our fast thinking response might be to say: "Well, those two effects are likely to cancel." This quick intuition can lead us astray. Looking at the functional dependence (how many powers) of what we are looking at, can help us see which of our two effects is more important. An example of how this works is given in the following problem.

An example: Implication of smoking on blood flow

Here's a sample problem that shows how paying attention to the functional dependence can give useful information about the relative strength of competing effects.

Edward R. Murrow,
public domain

Smoking tobacco is bad for your circulatory health. The nicotine from tobacco causes arteries to constrict. 

The resistance to flow of an artery (symbol $Z$) follows from the Hagen-Poisseuille equation and is given by

$$Z = \frac{8\mu L}{\pi R^4}$$

where $\mu$ is the viscosity of the blood, $L$, the length of the artery, and $R$ the radius of the artery.

If the radius decreases by 10%, can you overcome the effect on the resistance by taking a blood thinner to decrease the viscosity by 10%?

$Z$ is inversely proportional to the 4th power of $R$, so decreasing $R$ by 10% changes Z by a factor of $(1/0.9)^4 = 1.52$, an increase of more than 50%! Since $Z$ is directly proportional to the 1st power of $\mu$, decreasing $\mu$ by 10% will only reduce $Z$ by 10%. The combined effect will be a factor of $1.52 \times 0.9 = 1.37.$ The two 10% effects do NOT cancel because of their different functional dependences.

Joe Redish 7/8/11 and 5/15/19


Associated Problems

Article 273
Last Modified: December 24, 2021