Interacting systems

We now examine interacting systems. We will find that the number of states for the combined system is extremely sensitive to the distribution of energy among the interacting subsystems, having a very sharp narrow peak at some “optimum” value (Figure 7.1). Configurations corresponding to a greater number of accessible states are correspondingly more probable, so the distribution of energy is most probably at the peaked optimum value. Even the slightest deviation would cause a dramatic reduction in the number of accessible states and would therefore be very improbable.

This chapter is devoted to developing this statement of probabilities, which underlies the most powerful tools of thermodynamics. We elevate it to the stature of a “law.” Even though there is some small probability that the law may be broken, it is so minuscule that we can rest assured that we will never see it violated by any macroscopic system. Rivers will flow uphill and things will freeze in a fire if the law is broken. No one has ever seen it happen, and you can bet that you won't either.

Microscopic examples

We now investigate some examples of how the number of states is affected by the distribution of energy between interacting systems. Consider the situation of Figure 7.1, where an isolated system A0 is composed of two subsystems, A1 and A2, which may be interacting in any manner.

The general idea

Engines convert heat into work. Thermodynamics owes both its name, “heat-motion”, and much of its early development to the study of engines. The working system for most engines interacts both thermally and mechanically with other systems, so its properties depend on two independent variables. Most engines are cyclical, so that the working system goes through the following stages:

• it is heated;

• it expands and does work, pushing a piston or turbine blades;

• it is cooled further;

• it is compressed back into its original state, ready to begin the cycle again.

The expansion occurs when the working system is hot and is under high pressure or has a larger volume, and the compression occurs when it is cooler and is under lower pressure or has a smaller volume. Therefore, the work done by the engine while expanding is greater than the work done on the engine while being compressed. So there is a net output of work by the engine during each cycle. This is what makes engines useful. If you can understand this paragraph, you understand nearly all engines.

The details vary from one engine to the next. The working system could be any of a large variety of gases or volatile liquids. The source of heating could be such things as a flame, a chemical explosion, heating coils, steam pipes, sunlight, or a nuclear reactor. The cooling could be provided by such things as air, water, ice, evaporation, or radiative coils.

Overview

The preceding chapters introduced the fundamental ideas that connect the microscopic and macroscopic behavior of systems. They also gave an overview of the three types of interactions between systems and how the second law controls them. These concepts form the statistical basis of thermodynamics, and the tools are so general that they can be applied to almost any system imaginable. This is the single most impressive feature of the subject. Unfortunately, it is also the single most confusing feature of the subject. There are so many different kinds of systems and such a variety of parameters – internal energy, temperature, pressure, entropy, volume, chemical potential, number of particles, and many more. Furthermore, the interdependence among these parameters varies from one system to the next and in ways that are usually not specified. Consequently, we often deal with general and abstract expressions, each involving many parameters whose interrelationships are either vague or unknown.

But the large number of parameters can be turned to our advantage. We don't need them all, so we can choose to use whichever we wish and ignore the rest. Furthermore, their behaviors and interrelationships are heavily constrained. In this and the following chapters we learn how to make order out of chaos through a judicious choice of parameters and the application of constraints.

In this chapter we return to the study of systems that are nearly degenerate. As illustrated in Figure 24.1, a degenerate system of N identical fermions fills the N lowest quantum states, one particle per state. For degenerate bosons, all are in the one single state of lowest energy.

A degenerate system is confined to a small volume in phase space because of either:

• restricted volume in momentum space owing to small masses, or low temperatures, or

• Restricted volume in coordinate space owing to high densities.

Important examples of each case will be studied in this chapter.

The measures of high densities, small masses, and low temperatures are on a relative scale and are interdependent. At earthly densities, many systems become degenerate only if temperatures descend to a few kelvins or lower. Yet these same systems may be degenerate at several million kelvins when squashed together in extremely dense collapsed stars. The conduction electrons in metals have the same density and temperature as the atoms. Yet because of the difference in their masses, the electrons are degenerate and the atoms are not.

At high temperatures and/or low densities, the particles of a system are surrounded by vacant quantum states into which they can move (Figure 24.1). This freedom enables them to give rather smooth and continuous responses to varying environmental conditions. But in degenerate systems, quantum effects are more visible. Imprisoned particles are unable to move into neighboring states, which makes the system unresponsive to external stimuli.