2 - Statistics for small systems

Summary

As indicated in Chapter 1, we will begin our studies by considering “small systems” – those with relatively few elements. Small systems are important in many fields, such as microelectronics, thin films, surface coatings, and materials at low temperatures. The elements of small systems may be impurities in semiconductors, signal carriers, vortices in liquids, vibrational excitations in solids, elements in computer circuits, etc. We may wish to study some behavioral characteristic of a small population of plants or people or to analyze the results of a small number of identical experiments. Besides being important in their own right, the pedagogical reason for studying small, easily comprehensible systems first is that we gain better insight into the behaviors of larger systems and better appreciation for the statistical tools we must develop to study them.

The introduction to larger systems will begin in Chapter 4. Each macroscopic system contains a very large number of microscopic elements. A glass of water has more than 10²⁴ identical water molecules, and the room you are in probably has over 10²⁷ identical nitrogen molecules and one quarter that number of identical molecules of oxygen. The properties of large systems are very predictable, even though the behavior of any individual element is not (Figure 2.1). This predictability allows us to use rather elegant and streamlined statistical tools in analyzing them.

By contrast, the behaviors of smaller systems are more erratic and unpredictable, requiring the use of more detailed statistical tools.