Partitioning scales in physics

A physics theory is a mathematical construction used to predict the outcome of measurements and observations. Either the predictions are within the error margin of the measurement, or they are not. If they are, the theory “survives” the test of experiment. If they are not, as long as one trusts the good quality of the measurements, the theory is invalidated. Then, we must either discard the theory, or include finer effects to account for the difference between the original theory and the measurements.

The role of experimental errors is crucial, making experimental results essentially useless when they are missing. Despite the well recognized importance of experimental errors, they are not included at a fundamental level in the development of theories. By “experimental error”, we mean equivalently resolutions, or precisions of measurements.

The way we usually consider errors leads us to think that smaller errors or finer resolutions are systematically better. This is however not always the case. Indeed, there are situations in which finer resolutions do not provide “better” results but simply different results. The example of the length of the pencil in our previous post already hinted at this. Another historical example is when physicists of the late nineteenth and early twentieth century started probing the atomic and subatomic world. They quickly realized that physics theories of the time could not account for the new observations. Neils Bohr, despite the success of his atomic model, understood that classical physics could not explain how and why the hydrogen atom composed of an electron “orbiting” a proton could be stable.

A new physics, quantum mechanics, had to be developed. It was quickly formulated as a collection of mathematical axioms, whose justification stems from the astounding predictive power they provide. However, these axioms are generally not considered to derive from any more fundamental principles. The resulting quantum mechanics is so bizarre and counter-intuitive that it has fueled very heated and still ongoing scientific and philosophical debates. In quantum mechanics, observations at different scales change the nature of what is observed and we will discuss this later on.

Other examples where different resolutions reveal different aspects arise in complex systems. Complex systems have constituents or mechanisms operating at different scales, which all play an important role in the structures, behaviours or functionalities of the system as a whole. Branches of science dealing with complex systems are various, from astrophysical structure formation, geology, meteorology to biology, ecology, sociology or economy. These domains are interrelated and are further subdivided into more specialized branches of research, often corresponding to different scales.

In physics we generally proceed by approximation, looking for ways to neglect some less relevant aspects of the system. Often, this is done by assuming -more or less explicitly- that mechanisms or processes operating at a given scale may not have much of an influence at other scales. For example, when studying the mechanics of a spring, we can safely ignore that Earth is spherical, or that the Earth is orbiting the Sun. In doing so we take advantage of the fact that the considered spring is typically much smaller than the Earth and the Solar System. Similarly, we know very well that fluids are composed of atoms and molecules but we do not give much consideration to this fact in fluid mechanics, where we describe fluids using density, temperature and pressure fields.

In physics, we always try to take advantage of the fact that, in a first approximation, different scales are essentially decoupled from each others. Without this type of simplifications, we may still be able to write physics equations, but they typically become intractable and useless for understanding what is going on. This is why in physics, we generally do not have much to say about complex systems, where the method of scale partitioning breaks down. Indeed there is no general physics theory or even method to deal with complex or chaotic systems. We will come back more closely to the case of chaotic systems.

Figure 1: In Jurassic Park, obnoxious Ian Malcom (Jeff Goldblum) gives a quite good description and demonstration of the unpredictability of complex and chaotic systems.

The principle of relativity requires the laws of physics to be valid from the viewpoint of any reference frame, independently of their relative positions, orientations and motions. The enforcement of this relativity principle however fails to induce quantum mechanics. It also fails to induce any usable theory for complex or chaotic systems. Scale relativity proposes to consider and deal with relative resolution scales in the same way as relativity theories deal with positions, orientations and motions. We may thus expect scale relativity to provide a new and different approach to both the quantum and complex domains. This will be a core topic of discussion in this blog. In fact, there are geometrical objects, fractals, that have scaling properties similar to quantum mechanical and complex systems behavior.