Beyond Calculus Based Physics

Physics is typically concerned with the discover and categorization of the fundamental particles (players) and forces (interactions).

For the most part, this has been limited to forces that can be described using calculus, ypically in one of the following forms:

$$\vec{F} = \frac{\mathrm{d}\vec{p}}{\mathrm{d}t}$$
$$\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial L}{\partial \dot{q}}\right) - \frac{\partial L}{\partial q} = 0$$

These are often called the equations of motion for a system.

While this is the typical approach, what happens if there are "forces" that cant be described by calculus?

Consider a typical example of heat flow on a 2D plate. One way to describe the "force" is that the temperature of any site is the average of neighboring sites. This rule produces very real time dependent temperatures for the plate.

In this case, the heat flow can be represented using calculus.

$$\frac{\partial T}{\partial t} - k\nabla^2 T =0$$

Part of the reason why it is possible to use calculus is because the process has a characteristic length/time scale governed by k.

What happens when the phenomenon has no characteristic length scale as is the case with forest fires, earthquakes, avalanches, river networks, epidemics, turbulence, etc.

There is typically not a straightforward way of writing a prescriptive dynamic using calculus, and there is no guarantee that you you can describe the "forces" in the system using calculus.

One example of this problem is the forest fire model.

Natural forest-fires are known to follow power-law frequency-area distributions.

Distribution of natural forest fires in different regions. Figure taken from Malamud et. al [1]
Forest Fire Distribution

The is really no clear way of writing the dynamic (ignition, burning, etc.) using calculus.

In the same way that Newton had to invent calculus to describe his ideas, perhaps we need a new form of math to describe more general forms of dynamics.

One question frequently asked when using these generalized dynamics is whether the dynamic represents a "real" force? or is the dynamic just a simple way of getting the results.

We can answer this question with another question: In electrodynamics, are the fields or the potentials real?

$$\nabla \cdot \vec{E} = \frac{\rho}{\varepsilon_{0}}$$
$$\nabla \cdot \vec{B} = 0$$
$$\nabla^2\varphi + \frac{\partial}{\partial t}\left(\nabla \cdot \vec{A} \right) = -\frac{\rho}{\varepsilon_{0}}$$
$$\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$$
$$ \left(\nabla^2\vec{A} - \mu_{0}\varepsilon_{0}\frac{\partial^2\vec{A}}{\partial t^2}\right) + \nabla\left(\nabla \cdot \vec{A} +\mu_{0}\varepsilon_{0}\frac{\partial \varphi}{\partial t}\right) = \mu_{0} \vec{J}$$
$$\nabla \times \vec{B} = \mu_{0} \left(\vec{J} +\varepsilon_{0} \frac{\partial \vec{E}}{\partial t}\right)$$

There are arguments against both. The potentials change arbitrarily with the choice of gauge and so give a non-unique reality. However, quantum mechanics depends on the potentials and not the fields as shown by experimental verification of the Ahronov_Bohm effect. So which is real?

Neither! Both are mathematical representations that can be used to make falsifiable predictions. Science can only distinguish between models that make different predictions given the same conditions. Part of this debate goes back to the difference between dynamics and kinematics.

In the end it does not matter. Physics/Science is all about describing how reality works not what is real. What is real is a question of philosophy.

Because prescriptive dynamics can be used to make falsifiable predictions (and have been show to be reliable) they are valuable models.

Currently we don't have a unified mathematical framework for describing prescriptive types of dynamics. Perhaps there will be a way to describe these dynamics using calculus? Perhaps we need some new mathematics?

In the mean time, we can explore these types of phenomenon to understand them in greater detail and to explore what types of behaviors any new mathematical framework would need to be able to represent.

These prescriptive dynamics do not fit the traditional definition of a force, but have been very successful in describing many natural phenomenon.

 

References

[1] Malamud, B. D., Morein, G., & Turcotte, D. L. (1998). Forest Fires: An Example of Self-Organized Critical Behavior. Science, 281(5384), 1840–1842. doi:10.1126/science.281.5384.1840