ago, but despite many related technical results, the basic reasons for its validity have until now remained rather mysterious.

The field of thermodynamics is generally concerned with issues of heat and energy in physical systems. A fundamental fact known since the mid-1800s is that heat is a form of energy associated with the random microscopic motions of large numbers of atoms or other particles.

One formulation of the Second Law then states that any energy associated with organized motions of such particles tends to degrade irreversibly into heat. And the pictures at the beginning of this section show essentially just such a phenomenon. Initially there are particles which move in a fairly regular and organized way. But as time goes on, the motion that occurs becomes progressively more random.

There are several details of the cellular automaton used above that differ from actual physical systems of the kind usually studied in thermodynamics. But at the cost of some additional technical complication, it is fairly straightforward to set up a more realistic system.

The pictures on the next two pages [446, 447] show a particular two-dimensional cellular automaton in which black squares representing particles move around and collide with each other, essentially like particles in an ideal gas. This cellular automaton shares with the cellular automaton at the beginning of the section the property of being reversible. But it also has the additional feature that in every collision the total number of particles in it remains unchanged. And since each particle can be thought of as having a certain energy, it follows that the total energy of the system is therefore conserved.

In the first case shown, the particles are taken to bounce around in an empty square box. And it turns out that in this particular case only very simple repetitive behavior is ever obtained. But almost any change destroys this simplicity.

And in the second case, for example, the presence of a small fixed obstacle leads to rapid randomization in the arrangement of particles—very much like the randomization we saw in the one-dimensional cellular automaton that we discussed earlier in this section.