Time-Dependent Solutions to Fick’s Equations

Solving the diffusion equation (also known as Fick’s second law) can require fancy mathematics. After discussing a few solutions in Intermediate Physics for Medicine and Biology, Russ Hobbie and I write

Many other solutions to the diffusion equation and techniques for solving it are known. See Crank (1975) and Carslaw and Jaeger (1959).

Random Walks in Biology,
by Howard Berg.

In Random Walks in Biology, Howard Berg addresses the same point.

Time-dependent solutions to Fick’s equations

One way to find solutions to Fick’s equations is to look them up! An excellent source is Carslaw and Jaeger (1959), a book dealing with the conduction of heat in solids. The heat equation has the same form as the diffusion equation [see Chapter 4, Homework Problem 19 in IPMB]. In the notation of Carslaw and Jaeger,

where ν is the temperature and κ is the thermal diffusivity. So, take their results and read C for ν and D for κ. Sources that do not require such translation include Crank (1975) and Jost (1960). But this strategy requires luck. If you happen to find a discussion of just the problem that you are trying to solve, well and good. If not, you will soon be lost in a morass of complex equations.

After reading this, my first thought was: I know Crank, and I know Carslaw and Jaeger, but who’s Jost? The reference is to

Jost, W. 1960. Diffusion in Solids, Liquids, Gases. Academic.

The Oakland University library does not own this book, but even if it did the book might as well be on the moon for all the access I have to it. I can still download journal articles through the library website, but the library building itself is locked up tight because of the coronavirus.

If Jost’s book is anything like Crank’s or Carslaw and Jaeger’s, it’s overflowing with mathematics (just the way I like it).

I was able to find an obituary for Wilhelm Jost, written by Hartweil Calcote.

F. Wilhelm Jost

1903-1988

Professor Dr. sc. nat. Dres. h. c. W. Jost died on September 25, 1988 in G6ttingen. W. Jost was a preeminent scientist who pioneered research and development in combustion, reaction-kinetics in gases and solids, diffusion in solids and phase separation. He authored several monographs that have become classics in the field, among them “Explosions- und Verbrennungsvorg/inge in Gasen” with English and Russian editions. Several Academies of Science elected him as a member and he served on the boards of many scientific societies.

W. Jost was a director of the International Combustion Institute for decades and he founded and directed its German section. His many distinguished awards include the Institute's Sir Alfred Egerton Gold Medal.

F. W. Jost was born in Friedberg in Hessen. He studied chemistry in Halle and Munich and received his Dr. sc. nat. degree in chemistry from the University of Halle in 1926. He then joined M. Bodenstein in Berlin where his work on gas kinetics and hydrocarbon oxidation started. In 1929 Jost became Privatdozent in Hannover. From 1932 to 1933 he spent a year at MIT in Cambridge, Massachusetts, where he founded the basis for the understanding of disorder energies. In 1937 W. lost became a Professor of Physical Chemistry in Leipzig, in 1943 at the University of Marburg and 1951 in Darmstadt, in 1952 he accepted a call to the chair of W. Nernst at the University in G6ttingen.

On the occasion of the honorary promotion in Cambridge it was stated: "This man has always shown himself zealous for liberty, careful of truth and thoroughly civilized".

The members of the International Combustion Institute express their deep sympathy to his wife and his family.

Solving a differential equation by looking the solution up is an odd way to do math, but for the diffusion equation it often works. I will add Jost’s book to my list of post-pandemic reading.