Hot molecules, cold electrons

Paul Nahin is well known as a popular science writer. Some twenty books he has published since he started at the end of the previous century with a biography of Oliver Heaviside. Most of his books are dealing with topics involving physics, but there is always keen attention given to mathematics. For example he authored books on explicit mathematical topics like An imaginary tale. The story of √-1 (1998) and Dr. Euler's fabulous formula (2006).

The present book is like a mathematical textbook for engineering or science students in which all the derivations are given. Nahin uses an historical approach to introduce Fourier analysis, derive the heat equation, and solve it for different geometries and boundary conditions. When applied to a cooling sphere, this illustrates how William Thomson (Lord Kelvin) estimated the age of our planet by computing how a molten sphere cools down to a sphere with a solid crust (that explains the hot molecules of the title). When the equation is solved for a long cable, it explains how electrons travel through the transatlantic submarine telegraph cable (hence the cold electrons).

So there are a lot of formulas and derivations, but it is not a course as it would be written in modern times. It is taken out of a regular university curriculum and it assumes only the basic calculus from a course at a first year science, engineering, or mathematics level. Fourier series and the Fourier transform are developed from basic principles. Nowadays, the heat equation can be solved efficiently using for example Laplace transforms, but Nahin prefers to use essentially the mathematics available to Fourier who solved it in the time domain. Every step is explained to the smallest details. Sometimes the approach is using an engineering style of mathematics. This means that Nahin is just using an insight from the underlying physics to propose a certain method or to justify a certain solution. Infinite sums and integrals are interchanged, postponing to when the eventual result is obtained whether this makes sense or not. The square root of minus 1 is however denoted by the mathematical standard i, and not by j as is customized by the engineering community to distinguish it from electric current which is also indicated by i or I.

This "engineering mathematics" is also what Fourier applied. His original report on the solution of the heat equation in 1807 was criticized by Lagrange and Laplace because he used his formally obtained infinite sums as if they were ordinary functions. It is not until his "new mathematics" was better understood, ten years later that he was taken seriously and was accepted as a member of the French Academy of Science. Chapter 1 is an eye opener to the sort of mathematics that Fourier introduced. It is for example shown how Fourier obtained $\frac{\pi}{4}=\sum_{k=0}^\infty (−1)^k\frac{\cos(2k+1)x}{2k+1}$. This is well known for $x=0$ (Leibniz formula), but there are many other values of $x$ for which this is also true, much to the surprise of Fourier's contemporaries.

In Chapter 2, the Fourier series are derived and it is shown that they are optimal approximations in a least squares sense. Convergence is not proved. Nahin asks the reader to "accept that our mathematician colleagues have, indeed, established its truth". In this way Fourier series, the Parseval identity, Dirichlet's integral, and the Fourier transform are introduced.

Chapter 3 derives the heat equation $\frac{\partial u}{\partial t}=k(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}+\frac{\partial^2 u}{\partial z^2})$ from first principles. When the medium is a long radiating cable, it is essentially one dimensional and a simple solution is found as a decaying exponential assuming a constant energy loss per unit length, not depending on time. The solution of the equation for different geometries and different physical boundary conditions is discussed in the next chapter. It starts with a cooling problem of an infinite slab with finite thickness ($0\le x \le L$) using a separation of variables ($u(t,x)=f(t)g(x)$) as Johann Bernoulli did. This results in an infinite series with terms of the form $\exp(−ak^2t)\sin(bkx)$ which has to satisfy the boundary conditions. Next, the spherical problem is solved. Assuming isotropy for a sphere, it becomes one-dimensional in the radius $r$. This problem was solved by Lord Kelvin when he applied it to a cooling Earth, which however drastically underestimated its existence to 98 million years because he did not know about radioactive decay or tectonic plates. Next is the solution in a semi-infinite medium with infinite thickness. This is the first case of the slab where the thickness $L$ goes to $\infty$. This is an occasion to show how the Fourier series used for finite $L$ migrates into the Fourier transform when $L\to\infty$. The heat equation is also solved for other cases like a circular ring and an insulated sphere These were also discussed by Fourier in his Théorie analytique de la chaleur (1822), although the last one did not result in a Fourier series.

Chapter 5 starts with a crash course on electrical circuits: resistors, capacitors, inductors and Kirchoff's laws and describing the behaviour of electrons in an electrical field. And lo and behold, the electrons in a one-dimensional semi-infinite induction-free telegraph cable behave according to the heat equation, again an ingenious insight of Lord Kelvin. Solving that equation was a theoretical achievement, producing the cable and letting it sink to the bottom of the ocean was a risky and adventurous enterprise. In this book, that technological adventure is only lingering in the background. A nice account of this adventure can be found for example in the book A Mind at Play: How Claude Shannon Invented the Information Age by J. Soni and R. Goodman (2017).

Heaviside also features in the last chapter discussing the evolution after the 1866 Atlantic cable was realized. He added the inductance to the heat equation which turns it into a wave equation (actually the telegrapher's equation describing traveling waves in transmission lines, smartly solved by d'Alembert). That removes the assumed instantaneous action at a distance in the heat equation, which was causing a diffusion of the signal. The parameters of the cable can be controlled to remove that effect and this improved the usefulness of the cable considerably. Nahin ends by discussing the computation of how an arbitrary signal is transmitted. The diffusion however destroys the information during the transmission. This is illustrated by a matlab program that computes this deformation. The short code is given so that you can try it out yourself. The example shows that the signal is unrecognizable, it can still work though for a binary signal since the only information that one needs to detect is whether or not a bit is zero or one. We can also read how Heaviside explained the asymmetry of the transmission time: a message sent from England took longer than a message sent to England.

The sources used by Nahin, and some additional historical notes are listed at the end of the book, organized per chapter. There is no separate bibliography but there is an index that includes references to these notes. He has also one appendix about Leibniz's formula, i.e., how to compute the derivative of an integral if the boundaries of the integral are varying.

The book confirms what is already known from his previous books: Nahin knows how to write a book mixing physics and (a lot of) mathematics and (still) make it readable for a (relatively) broad public (with only some basic mathematical knowledge). The mathematics in this book certainly take the leading role like it does in lecture notes about the solution of differential equations. Nahin takes his time to explain everything and derive things from the very basics. When the mathematics become too involved or advanced, he uses intuition and asks the reader to accept and believe the result. The hard core mathematical mind may have some problems with his "engineering approach", but it works perfectly well for a first introduction. Anyway, from the historical perspective, this approach was used by the people who originally developed the theory.

Adhemar Bultheel
Book details

Nahin introduces us through an historical approach to Fourier series, Fourier transforms, and how Fourier used this to solve the heat equation. Lord Kelvin used the heat equation to model the cooling of the Earth and hence estimate its age and he, and others, solved essentially the same equation to model the flow of electrons in the transatlantic telegraph cable.

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9780691191720 (hbk)
USD 24.95 (hbk)