The fun with the dice (well, I enjoyed the counting) shows nicely how energy can be distributed but it gives an incomplete picture of the message of Maxwell-Boltzmann statistics. A central issue is how microstates are distributed in phase space, in particular that the equilibrium configuration will fluctuate around the most probable distributions. All other distributions, even those with the right energy, are unlikely to occur because there are so many more distributions close to the most probable ones. This isn't obvious with a small number of particles but in a thimblefull of atmosphere there are over 10¹⁹ molecules and then it is very true. Suppose the dice are re-interpreted to represent the number of molecules in a given cell in phase space. Each spot on a dice now represents a molecule. The total number of molecules is constant and hence (as before) the changes taking place with time must preserve the sum of the spots. In Maxwell-Boltzmann statistics all the particles are distinguishable and hence we have to count the spots as distinguishable, say of they are of different colours¹. It was Boltzmann who showed that the probability of a configuration was proportional to the sum of the logarithms of the occupation numbers of each cell (i.e. the numbers showing on the dice). This is identified as a measure of the entropy of the system. A gas not in equilibrium will evolve to maximise its entropy just because this represents the maximum probability among the possible states of its constituents. The motivation for both Maxwell and Boltzmann was to understand gases but it later became apparent that their ideas were very widely applicable.

Imagine that a gas is let into an almost empty chamber through a nozzle at one side. It will quickly spread out towards the most probable distribution, which is one uniformly filling the chamber. On a slower timescale, think of a drop of ink allowed to fall into a beaker of water. Much more slowly than a gas moves, the ink will diffuse throughout the beaker to take up its equilibrium configuration. If you could collect statistics on the location and speed of ink molecules in the water you would be able to predict that in future the odds of the ink congregating in one part of the beaker are almost infinitesimally small. The ink will stay diffused throughout the beaker. Seeing the ink all through the beaker, you could not deduce that at some time in the past the ink was altogether in one small drop just below the surface and even if you thought it might have been you can't deduce how long in the past that was. This is what Maxwell meant by saying that the future distribution of matter in motion is more predictable than its past history.

Returning to the dice representing energies, of course 6 available energies is a gross simplification. Moreover, in the realm of discrete energies and atomic or molecular objects quantum mechanics rules. One important difference is that in quantum mechanics, particles of the same kind are indistinguishable. The main reason for the large number of possibilities in arranging the dice in the previous example is that every die in the row has been taken as distinguishable from every other one. For example, there are 10!/(4!4!) (= 6300) ways of rearranging the same set of dice as are illustrated. In quantum mechanics that arrangement would be counted only once. Quantum particles obey either Fermi-Dirac or Bose-Einstein statistics. In spite of the apparent gross overcounting, Maxwell-Boltzmann statistics are a good approximation in many circumstances even though they don't strictly apply at an atomic level. That was found out in the 20th century, long after either Maxwell's or Boltzmann's time.

¹footnote: This footnote might help in understanding the reason why a large number of cells concentrates the system near the most likely configuration.  Suppose there are N = 10¹⁹ cells in a small volume containing 2×10¹⁹ molecules.  The microstate that has the most number of possibilities of occurring is the one where the molecules are uniformly distributed throughout the volume, i.e. there are 2 molecules in each cell.  The number of possibilities P is N!/2!^N.  Call this P.  Now suppose just 1% of the cells change to an occupation of 3 and 1% to an occupation of 1 (to preserve the total number of molecules).  The number of possible microstates is now N!/(2!^0.98N*3!^0.01N*1!^0.01N).  Call this P’.  The ratio of P’/P after a couple of lines = (2/3)^0.01N = e^{0.01N*ln(2/3)} ≈ e^-0.004N.  Given that N = 10¹⁹, the ratio of P’/P is incredibly small so even this minor fluctuation from equilibrium is really unlikely to happen, for the number of microstates available with this slightly non-uniform distribution is comparatively tiny.  

There is one redeeming feature in that the mean time between molecular collisions under normal atmospheric conditions is around 10^-10 seconds so the 10¹⁹ dice are thrown about this often.   In 1860 Maxwell introduced the concept of the number of collisions per second experienced by a typical molecule and found it was about 10¹⁰.  He was ‘in the right ball-park’.  Even a 1-in-a-billion fluctuation from the mean will very fleetingly occur a few times a second, meaning that the density in every small volume of the gas fluctuates.  This phenomenon is responsible for the twinkling of stars and, more subtly, for the blue colour of a clear sky.

JSR 2016