Thomas Enda Conlon

Physics and Statistics

Generally speaking, making a connection between topics in maths and physics which have been taught completely separately is a ”getting two for the price of one ” event for a student and the connection embeds both topics more firmly in his or her mind . If the connection is a particularly insghtful one – like that between the tangents to curves and the areas under them – then the benefit is of course spectacularly more than just two for the price of one.

Since statistics recently became a mandatory component of A Level maths and some kinetic theory of gases has always been on the Physics syllabus, there is a relatively new opportunity to make a connection between these topics which I think should be taken. This is that in an ideal gas of identical particles of mass m in equilibrium the the components of the velocity in a given direction (e.g. the x – direction) are normally distributed with mean 0 and variance kT/m where k is Boltzmann’s constant and T the temperature in Kelvin. As the x,y,z directions are mutually independent this leads quickly, using A Level stats, to the mean kinetic energy depending only on the temperature. Also Maxwell in Proposition VI of his famous 1859 paper gives an argument for the equalisation through collisions of the mean kinetic energies of particles of different masses. The argument is difficult to follow and I have attempted to clarify it here.

When I was a student I learned to integrate ln x by the trick of treating it as 1 x ln x and integrating by parts. I have since realised that it is much more geometrically insightful to use the fact that the graph of y = ln x is the mirror image of y = exp(x) through the line y = x. This allows us to change the problem of integrating ln x between bounds to integrating exp(x) between (different) bounds. The general lesson is that if we can integrate a function we should expect to be able to integrate its inverse. See details here.

Another paragraph, See details here