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Unfortunately, one of my favourite concepts in engineering—in which I have a degree—appeared in my last term of studies: >statistical mechanics . I definitely wouldn't have been the best at it but I'm sure I would've enjoyed the course more if it had appeared earlier in the curriculum.
Like the name suggests, statistical mechanics uses concepts from statistics to solve problems in mechanics. The concepts become particularly relevant when the mechanical system contains a large number of smaller particles, like molecules of a gas. Depending on how individual molecules behave, statistical mechanics can tell us how a gas of such molecules will behave in different conditions.
Statistical mechanics is a subset of ergodic theory. Given a system, like the gas, ergodic theory contains the mathematical tools necessary to predict how the gas can behave in all possible conditions. In terms of the gas itself, put another way, ergodic theory lets us calculate how the gas explores its different states. Such a set of tools is powerful when we study a system that seems to behave randomly but whose constituents are actually following certain well-defined rules.
This year’s >Abel Prize —considered to be as prestigious as a Nobel Prize—was awarded to an American mathematician named >Yakov Sinai in recognition of his development of tools in ergodic theory. Sinai, of Russian origin, currently works in the Institute of Advanced Studies, Princeton University. His PhD adviser, at Moscow State University, was none other than >Andrey Kolmogorov , one of the greatest mathematicians of the 20th century.
The solutions forwarded by Sinai have found applications in diverse fields. One such practical system similar to the example of the “random gas” that comes to mind is climate change. Other fields include the study of planetary motion, population growth and >communication theory . One of the more accessible examples where Sinai’s work has proved a difference is depicted below.
Image: Wolfram Mathworld
A forest of cylindrical mirrors of two sizes are arranged in a repeating pattern as shown above. When a ray of light is injected into this forest, its path through the maze is can take >a variety of shapes depending on the angle and place at which it was injected. In fact, Sinai’s work lets us understand that even with a small change in initial conditions, the path of the light can vary drastically.
In the study of dynamical systems, such an overt dependence on initial conditions is colloquially summarized as the >butterfly effect : a butterfly’s fluttering of its wings in one part of the world causing a tornado in another. A more entertaining depiction of this ‘effect’ is in time-travel: if you travelled to the past and changed something, the world of today would be drastically different.
Sinai himself devised an interesting scenario to illustrate his work in ergodic theory (at the time), called Sinai billiards. Imagine a square box at the centre of which is a circular wall. Drop a ball inside such that it bounces off the walls. Assuming that the ball doesn’t lose energy and for a particular gravitational strength, Sinai proved that the ball’s path in the box will traverse all the space available given enough time.
>A simulation of Sinai billiards in action
In effect, Sinai’s work has bridged two worlds: the probabilistic world, where stuff probably happens, and the deterministic world, where we know for sure if something’s going to happen or not. Therefore, his contributions help mathematicians, engineers and physicists work out solutions to larger, more complex problems based on the principles that govern smaller, simpler objects.
