Mathematica

Now here's a neat idea from Mr Wolfram. The Computable Document Format. So I ( with Mathematica ) can show you ( possibly without Mathematica ) what I have produced. Download the free Wolfram CDF Player. Now you can have a look at this, which is a plot of the famous, classical and beautiful, Boltzmann distribution for a collection of particles in equilibrium !!

The horizontal axes are temperature ( 0 to 1000 from front to back and with a color gradient ) and energy level ( from left to right ). The vertical scale is state population relative to the lowest. So at zero Kelvin all the entities are huddled at the lowest energy levels and if the temperature is increased higher energy levels gain more population ( so the surface rises towards the back ). Don't take the energy axis units too seriously as I have effectively scaled to make Boltzmann's constant equal to one. The mesh lines parallel to the energy axis give successive exponential curves and illustrates how increasing temperature reduces the decay of that with temperature, thus embodies :

Exp[-E/kT]

I will explore the idea of these CDF's having some controls embedded in them, sliders and whatnot, that vary some parameter(s) and thus animate or demonstrate some mathematical features more fluently. I think this is a brilliant concept.

Cheers, Mike.

( edit ) There's an interesting codicil to this. The strict thermodynamic definition of temperature ( see the Zeroeth Law ) requires consideration of systems at equilibrium. Boltzmann's distribution, at least classically, describes such systems and so one can actually use the pattern of populations to define thermodynamic temperature. Thus IF a system is in equilibrium AND only the lowest energy states are occupied THEN the temperature is low.

Boltzmann's distribution assumes that the entities within are distinguishable. Classically this is possible, for while we can't 'paint' particles we can follow their tracks ( gedanken experiment here ) by arbitrarily and thus infinitesimally low energy probes ( eg. photons ) and thus 'individualise' our description. So we could characterise our microstates in detail while remaining (a) infinitesimally close to an isolated system and (b) keep the disturbance to the particles also arbitrarily low and hence the distribution is as close to a pure Boltzmann shape as we like.

However using quantum mechanics, whatever probe we use will not simultaneously have an arbitrarily low energy and an arbitrary resolution ( Heisenberg ). So we must use the statistics of indistinguishable particle collections, either Fermi/Dirac or Bose/Einstein. In the limit of either high energy and/or low density all these statistical patterns approach each other.

( edit ) Conversely IF a system is at equilibrium AND has all energy states equally occupied THEN the temperature is infinite ! :-)

I agree, I was doing serious stuff within the hour. :-)

Cheers, Mike.