One might nevertheless imagine that it would be possible to devise a complicated machine, perhaps with an elaborate arrangement of paddles, that would still be able to extract systematic mechanical work even from an apparently random distribution of particles. But it turns out that in order to do this the machine would effectively have to be able to predict where every particle would be at every step in time.

And as we shall discuss in Chapter 12, this would mean that the machine would have to perform computations that are as sophisticated as those that correspond to the actual evolution of the system itself. The result is that in practice it is never possible to build perpetual motion machines that continually take energy in the form of heat—or randomized particle motions—and convert it into useful mechanical work.

The impossibility of such perpetual motion machines is one common statement of the Second Law of Thermodynamics. Another is that a quantity known as entropy tends to increase with time.

Entropy is defined as the amount of information about a system that is still unknown after one has made a certain set of measurements on the system. The specific value of the entropy will depend on what measurements one makes, but the content of the Second Law is that if one repeats the same measurements at different times, then the entropy deduced from them will tend to increase with time.

If one managed to find the positions and properties of all the particles in the system, then no information about the system would remain unknown, and the entropy of the system would just be zero. But in a practical experiment, one cannot expect to be able to make anything like such complete measurements.

And more realistically, the measurements one makes might for example give the total numbers of particles in certain regions inside the box. There are then a large number of possible detailed arrangements of particles that are all consistent with the results of such measurements. The entropy is defined as the amount of additional information that would be needed in order to pick out the specific arrangement that actually occurs.

We will discuss in more detail in Chapter 10 the notion of amount of information. But here we can imagine numbering all the possible arrangements of particles that are consistent with the results of our