1
Introduction

Before jumping right into everything, I want to spend a brief moment discussing statistical mechanics and thermodynamics. Why do we need these subjects and what are they? How do they differ and how do they connect with each other? Of course, you can jump to Chapter 2 and start learning the necessary material, but this will be a useful overview to frame our discussion over the next several hundred pages.

For most people, including undergraduate physics students, statistical mechanics tends to be one of the less well‐known subjects in physics. Every student knows about classical mechanics (even if not by that name when they first start studying physics), electricity and magnetism, and even quantum mechanics. Additionally, these subjects (along with more “exciting ideas” such as general relativity) are known by many non‐physics students as well. However, statistical mechanics is one of the “core subjects” that all physicists and engineers should understand in depth along with classical mechanics, electricity and magnetism, and quantum mechanics.1

Students are of course familiar with thermodynamic concepts (heat, temperature, pressure, etc.), and while many schools offer a course with this title, it is also often called Statistical Physics, Statistical Mechanics and Thermodynamics, or some variation on these. When registering for this class, many times the only idea students have regarding the course is that it “is the hardest class you’ll ever take in college, it makes no sense, and you no longer will understand what temperature is.”2 The salient question is then: What is the relationship between the (slightly) more familiar thermodynamics and this unfamiliar statistical mechanics? The quick answer, to some degree, is that they are two different approaches one can take to understand the same subject. I plan to give an overview in this chapter on how they relate and what the difference is when we consider this subject from either point of view. The goal is to understand how they are connected, why we study one over the other, and most importantly, why we’re going to start with statistical mechanics before moving on to the study of thermodynamics. After I discuss the two, as well as the approach we will take in this book, we will finally get started on the actual subject.

1.1 What is Thermodynamics?

Considering the constituent words that make it up, thermodynamics is merely the study of the motion of heat. Coming from the Greek, thermodynamiki, or, ϑερμοδυναμική, is a combination of thermótita (ϑερμότητα) which means “heat” and dynamikí (δυναμική) which means “dynamics.” In the early nineteenth century, when this field began to develop, the focus was on heat engines: using heat as a form of energy that can be converted into work (the reverse is easy to do, doing work to create heat). That the study of heat engines, and thus thermodynamics, exploded during this time is not surprising given that this overlaps with the Industrial Revolution. A steam locomotive is an early example of a widely used heat engine: You use the steam produced by burning coal to turn gears (thereby turning the wheels of the train and propelling it forward). In the process, and as the field progressed, four laws of thermodynamics were postulated and these allowed the introduction and development of various concepts such as temperature, entropy, enthalpy, and so forth.3

Many of these concepts are familiar from an everyday perspective (like temperature), or from hearing about it colloquially (like entropy). Still others, like enthalpy, are not well known by those who never took a chemistry class. The question remains though: What do these quantities describe physically? The meanings of these ideas can often be muddled (or worse, misunderstood) when first studying thermodynamics. Additionally, we will see that while starting from a thermodynamic perspective is nice from a conceptual (and historical) point of view, it can be very limiting. Finally, it is only natural to ask where thermodynamics fits in with regard to other areas of physics.

As physics developed throughout the centuries, it was necessary to begin to divide it into different subfields we can categorize very roughly by the relevant length scales and speeds in a given problem. A depiction of this is shown in Figure 1.1. If the relevant length scale of our system is denoted as , and the relevant speed is denoted as , then we can divide physics up as follows. Starting with classical mechanics (“the original physics”), and take on “everyday values”: values that are common for buildings, cars, people, etc. I’ll admit that the term “everyday values” is not a great way to describe these quantities, as classical physics is valid for a fairly wide range of sizes and speeds. The motion of dust ( m) as well as that of planetary orbits (Neptune is an average distance of about m from the sun), both can be described classically. Additionally, planets orbit the sun at high speeds (from our perspective) and they can be described classically (Mercury travels around the sun at almost m/s = 180 000 km/h). The precise length or speed scale is not important here; it just matters that there is a range such that the classical description is valid, and at some point it breaks down.

Figure 1.1 Various subfields of physics in relation to each other, when considering a range of length scales and speeds of the system under consideration.

Around the turn of the twentieth century, experiments were performed at much smaller length scales, and classical mechanics began to fail. Enter quantum mechanics, which becomes relevant as one nears molecular scales, so and smaller.4 Around the same time, the theory of special relativity was being developed, which is important to consider when our system approaches high speeds, specifically close to the speed of light . In each of these cases, we consider one of these quantities to change: In ordinary (non‐relativistic) quantum mechanics, speeds are not too high, while in special relativity, the length scales are those of classical mechanics, all shown in Figure 1.1.

Electricity and magnetism, the third of our four “core” subjects, doesn’t quite fit into any of these categories, so I added it to the middle of our plot, somewhat spilling over into various other subfields. It belongs a bit more in the special relativity section (as it was important in the development of this theory and in fact is already relativistic by nature), but often can be thought of as a classical topic. One thing to note for clarity though, while it overlaps various categories, it is specifically not quantum mechanical.

A couple of other subjects are thrown into the plot (just for some level of completeness) that aren’t always studied in an undergraduate curriculum. For large systems, general relativity (the fundamental theory of gravity) takes over for classical mechanics (in the figure, I have infinity in quotes to imply this is for very large systems without needing to specify an actual scale).5 For very small and fast systems, we combine quantum mechanics and special relativity to formulate quantum field theory (needed to study particle physics, including a quantum description of the electromagnetic field as well as the strong and weak nuclear interactions). While the use of a single length (or speed) scale is overly simplistic, systems with multiple scales that are wildly different are quite difficult to study in physics; this is one of the reasons why it is difficult to combine general relativity and quantum field theory (hence we have no quantum theory of gravity!).

All of the fields above are solvable directly in terms of laws that allow us to determine the equations of motion. For example,

• Classical mechanics: We can use Newton’s laws of motion to determine the trajectories of particles as long as we know all of the forces acting on them given a set of initial conditions.
• Electricity and magnetism: With Maxwell’s equations (and appropriate initial and boundary conditions), we can determine the electromagnetic field due to any charge and current distribution. Add in the Lorentz force, and we can describe (with classical mechanics) how charged particles move in electromagnetic fields.
• Quantum mechanics: We can solve the Schrödinger equation, along with appropriate initial and boundary conditions, to determine the wavefunction which can be used to calculate expectation values of physical observables.

While more complicated, the same can be said for the other advanced topics, as long as we have the relevant equations (and initial and/or boundary conditions). Keep in mind when we say we can solve these equations, this is all in principle: An exact solution is rarely possible in practice, while often approximate solutions are.

But what about thermodynamics, where do we put this in our figure? In Figure 1.1, I have implicitly assumed that the number of objects in our system is small, by which I mean one, two, or maybe three. A thermodynamic system, such as the air in the room you’re sitting in while reading this or the cup of coffee you are drinking to stay awake while...