Complexity_ A Guided Tour - Melanie Mitchell [24]
In short, classical mechanics attempts to say something about every single microscopic entity (e.g., molecule) by using Newton’s laws. Thermodynamics gives laws of macroscopic entities—heat, energy, and entropy—without acknowledging that any microscopic molecules are the source of these macroscopic entities. Statistical mechanics is a bridge between these two extremes, in that it explains how the behavior of the macroscopic entities arise from statistics of large ensembles of microscopic entities.
There is one problem with the statistical approach—it gives only the probable behavior of the system. For example, if all the air molecules in a room are flying around randomly, they are most likely to be spread out all over the room, and all of us will have enough air to breathe. This is what we predict and depend on, and it has never failed us yet. However, according to statistical mechanics, since the molecules are flying around randomly, there is some very small chance that at some point they will all fly over to the same corner at the same time. Then any person who happened to be in that corner would be crushed by the huge air pressure, and the rest of us would suffocate from lack of air. As far as I know, such an event has never happened in any room anywhere. However, there is nothing in Newton’s laws that says it can’t happen; it’s just incredibly unlikely. Boltzmann reasoned that if there are enough microscopic entities to average over, his statistical approach will give the right answer virtually all the time, and indeed, in practice it does so. But at the time Boltzmann was formulating his new science, the suggestion that a physical law could apply only “virtually all of the time” rather than exactly all of the time was repellent to many other scientists. Furthermore, Boltzmann’s insistence on the reality of microscopic entities such as molecules and atoms was also at odds with his colleagues. Some have speculated that the rejection of his ideas by most of his fellow scientists contributed to his suicide in 1906, at the age of 62. Only years after his death were his ideas generally accepted; he is now considered to be one of the most important scientists in history.
Microstates and Macrostates
Given a room full of air, at a given instant in time each molecule has a certain position and velocity, even if it is impossible to actually measure all of them. In statistical mechanics terminology, the particular collection of exact molecule positions and velocities at a given instant is called the microstate of the whole room at that instant. For a room full of air molecules randomly flying around, the most probable type of microstate at a given time is that the air molecules are spread uniformly around the room. The least probable type of microstate is that the air molecules are all clumped together as closely as possible in a single location, for example, the corner of the room. This seems simply obvious, but Boltzmann noted that the reason for this is that there are many more possible microstates of the system in which the air molecules are spread around uniformly than there are microstates in which they all are clumped together.
The situation is analogous to a slot machine with three rotating pictures (figure 3.2). Suppose each of the three pictures can come up “apple,” “orange,” “cherry,” “pear,” or “lemon.” Imagine you put in a quarter, and pull the handle to spin the pictures. It is much more likely that the pictures will all be different (i.e., you lose your money) than that the pictures will all be the same (i.e., you win a jackpot). Now imagine such a slot machine with fifty quadrillion pictures, and you