# Difference between phenomenological modeling and mathematical modeling

Is there a difference between phenomenological modeling and mathematical modeling? When reading a few journal papers, I often see the former being used -- is it just fancier wording? If it's relevant, the papers I'm reading have both modeling and lab experimental studies, and both the simulations and lab measurements are compared and analyzed.

A phenomenological model is based on observations of a system rather than on physical theory. Other physically based models are based on fundamental physical principles such as Newton's laws of motion. Both kinds of models might end up being expressed in the form of mathematical equations and called mathematical models.

In practice, models used in many areas of science and engineering combine both approaches to model development, with fundamental physical principles used where possible and empirical or phenomenological approaches used in parts of the model that can't be modeled from physical principles. These kinds of models are often called "semi-empirical models."

For example, you might want to model the motion of a simple pendulum and predict its period. By drawing a force diagram and accounting for the forces acting on the pendulum, you might derive a second-order ordinary differential equation model:

$$\theta''(t) + \frac{g}{L}\sin(\theta(t))=0$$

For small angles, $$\sin(\theta(t))$$ is approximately $$\theta(t)$$ so you might simplify your model to

$$\theta''(t)+\frac{g}{L} \theta(t)=0$$.

From this second differential equation, you could derive the solution

$$\theta(t)=\theta_{0}\cos(\sqrt{\frac{g}{L}} t)$$

and conclude that the period of oscillation is

$$T=2\pi \sqrt{\frac{L}{g}}$$.

The model that I've just described is based on physical principles, although it uses an approximation to simplify the equation.

In contrast to this, you could set up a bunch of experimental pendulums and observe how the period of the pendulum varies with $$L$$ and $$g$$ (set up a pendulum on Mars for example.) Once you've acquired enough data, you could guess (or better yet, use dimensional analysis to hypothesize) a model of the form

$$T \propto \sqrt{\frac{L}{g}}$$

and find the constant of proportionality by linear regression. The resulting model might be something like

$$T=6.3 \sqrt{\frac{L}{g}}$$.

This second model was built entirely from observations of the pendulum rather than being based on fundamental physical theory. This would be called a phenomenological or empirical model.

Both of the models described above are mathematical models in the sense that they're expressed in mathematical equations.

• Your 2nd paragraph, describing combining physical theory and using experimental data to arrive at a model, describes exactly what I've been studying, but the authors call it a phenomenological model, not a semi-empirical model. I guess it's not much of an issue? I'm just wondering whether I should follow their language, when I write my own paper. (Their papers are very well-known by many professors that I speak to, and the journal is top tier.) Apr 16 '20 at 16:09
• That will ultimately be an issue for you to settle with the referees and editor. Personally, I'd use "semi-empirical" to describe such a model if I were writing a paper, but I'd also be prepared to discuss using some other term with the editor. Apr 16 '20 at 16:43
• Of course many (perhaps most) real-life models are in between, combining fundamental theory and various empirical factors, scalings etc. where fundamental theory is not available. Apr 16 '20 at 19:46
• @MaximUmansky so, are you saying that using "phenomenological" in a journal paper is mostly fancier wording? Apr 16 '20 at 23:45
• @user35652 In my field (fusion plasma physics) there are theoretical models of two kinds, one is purely theoretical modeling, and the other is modeling incorporating some empirical relations; but both are usually referred to as theoretical modeling. On the other hand, in absence of theoretical models people sometimes use empirical scaling laws, and that could be called phenomenological modeling. But maybe in other fields the naming tradition is different. Apr 17 '20 at 15:27