In an earlier post I made the outrageous claim that three is a working approximation to infinity. If you thought that was ambitious, have a read of the following extract from an abstract that I discovered this morning while doing a bit of literature searching as part of my research. It’s a great insight into the mind of a theoretical physicist.
The present work studies the effect of additive noise on two high-dimensional systems. The first system under study is two-dimensional, evolves close to the deterministic stability threshold and exhibits an additive noise-induced shift of the control parameter when driving one variable by uncorrelated Gaussian noise…
Reference: Hutt, A. Additive noise may change the stability of nonlinear systems. Europhysics Letters, 84(3), 34003. (2008). DOI 10.1209/0295-5075/84/34003
Did you spot the implication? The author is implying that a two-dimensional system is high-dimensional. In other words, two is a big number. Now, I don’t know about you, but I live quite happily in three dimensions. This large number of dimensions doesn’t cause me any problems. But, when it comes to analyzing how systems behave, there is actually a massive increase in the diversity of behaviour when we move from a one-dimensional system to a two-dimensional system. (By a one-dimensional system, I mean one that needs just one variable to descibe it, and a two-dimensional system is one that needs two variables to describe it. In this sense a pendulum is a two-dimensional system. To describe its state you need to know the bob’s position and velocity.) Two dimensions are pretty diverse, really. The pendulum can sit still and just hang under gravity, or it can do its characteristic swing back and forth, but that’s not all. If you whirl it to start with it can go round and round in circles, or, if you start it pointing vertically upwards and ignore friction, it will drop (but which way?), swing round, and end up back exactly where it started.
The physicist, and perhaps more so the mathematician, will then make great strides forward, and happily move from two to three, four, ten and even infinite dimensional systems, which are just further examples of high dimensional systems. Visualizing what’s happening becomes a bit tricky, but, in practice, once you’ve got used to the idea, there is not a lot of difference between two and ten.