# Determinism, indeterminism and chaos

Physical phenomena in nature occur for some reasons, they follow specific patterns or laws, but can be predicted with a total certainty what the result will be?

Depending on the phenomenon we could predict the exact outcome or have an uncertainty. It is even possible that the only thing we get is a probable value according to a statistical criterion.

In the history of science we have gone through different stages. There was a time when all could be exactly predicted, being thus a deterministic period. However the discovery of new phenomena led to think that it was impossible to know the exact result of such phenomena, thus appearing a non-deterministic stream of scientific thinking. Later, the study of non-linear dynamical systems led to a new field of study: the study of systems with a completely erratic and unpredictable behaviour, though in principle, their formulation can be deterministic. This field is known as chaos.

The scientific determinism considers that, although the world is complex and unpredictable in many ways, it always evolves according to principles or rules that are totally determined, being chance something that only occurs apparently.

In the middle of the XIX century, determinism fell down piece by piece. There were two reasons for it.

Firstly, it was needed a completely and detailed knowledge of the initial conditions of the system under study to introduce them in the equations providing the system evolution and get a result.

Secondly, the dynamics of systems made of a large number of particles were very complex to solve.

This second reason was the one that made necessary to introduce concepts related to probability and statistics to solve the problems, giving as a result the creation of a new field of mechanics: statistical mechanics and thus a change in the scientific paradigm from a deterministic paradigm to a non-deterministic one.

The discovery of quantum mechanics had also consequences in the deterministic view of the world because from the Heisenberg’s uncertainty principle it is derived an impossibility to apply deterministic equations to the microscopic world because of the impossibility of knowing the value of two conjugate variables at the same time (e.g. position and speed)

In the mind of many people, it is associated the indeterminism with quantum mechanics and determinisms with classical physics, but, as it was demonstrated by Noble prize Max Born, determinism in classical mechanics is not real because it is not possible to establish with an infinite accuracy the initial conditions of an experiment.

On another hand, Feynman, in his famous lectures, said that the indeterminism does not exclusively belong to quantum mechanics but it is a basic property of many systems.

Almost all physical systems are dynamical systems, they are systems described by one or more variables that change with time.

There are dynamical systems that have a periodic behaviour ant other systems that do not have such behaviour. When the movement is not periodic, it depends on the initial conditions and it is unpredictable in long time intervals (although it can be predictable in short time intervals) it is said that the movement is chaotic.

In other words, chaos is a type of movement that can be described by equations, some times very easy, and that is characterised by:

• An irregular movement with time that has neither periodicities nor superposition of periodicities.
• It is unpredictable with time because it is very sensitive to the initial conditions.
• It is complex but ordered in the phase space

For example, when there are three different masses moving because of the action of gravity (let’s say three planets), the study of their evolution in time is really complex because their initial conditions, position and speed of the three masses. Poincaré showed that it was not possible to find an exact solution to the problem.

Lorenz atractor. Does it not resemble the wings of a butterfly (Source: Wikimedia Commons)

One of the most famous cases in the study of these non-linear dynamical systems took place in 1963 when Edward Lorenz developed a model with three ordinary differential equations to describe the movement of a fluid under the action of a temperature gradient (or in other words, he was studying the behaviour of the atmosphere) Using a computer he searched for numerical solutions to the system and found that it was very sensitive to the initial conditions. It was James York who recognised Lorenz’s work and introduced the term chaos.

Currently it is thought that after the discovery of quantum mechanics and Einstein’s relativity, all physics spins around these fields. However, chaos is a very wide field gaining followers not only physicists and mathematicians, but also in other fields such as biology, genetics and neuroscience. This interdisciplinary nature is amazing and shows how much some people can learn from others to make the science advance towards a greater knowledge of the world.

References

Las matemáticas y la física del Caos. Manuel de León, Miguel A. F. Sanjuán. CSIC

Caos. La creación de una ciencia. James Gleick.

E. Lorenz. Deterministic nonperiodic flow. Journal of the Atmospheric Sciences. Volume 20