In Japan, days from about April 29 until about May 5 are successive holidays as you know.

So today i.e., April 30, I could have enough time to read through a book entitled "Fourier Series" written by Georgi. Tolstov and translated into English by R.A.Silverman and published from Dover Publication. Reading former part was a teadious work because almost of it is the repeatition of "Theorem of something, The prooof of it", "The theorem of next thing, The proof of it,",,, and so on. But the last part of the book mentioned some physical applications of the Forurier theory, like string viblation, rectangular membrane viblation, and heat transfer in a cylinderical rod.

What has astounded me was a explanation about the heat propagation equation in an infinite length rod of which lateral surface was surrounded by thermal insulator. I would like to copy the part from the book that "formula(25.10)(which expesses the solution u(x,t)) shows that as time increases, u(x,t) approaches 0, i.e., the heat "spreads out" along the rod. On the other hand, (25.10) shows that the heat is "transmitted" instantaneously along the rod. In fact, let the initial temperature be positive for x0<=v<=x1 and zero outside this interval. Then it is clear that (an intgration equation is written and ) from which it is clear that u(x,t)>0 for arbitrarily small t>0 and arbitrarily large x."

This portion is explaning what happens in the temperature distribution u(x,t) along the infinite length rod after setting an initial condition of rectanguler temperature distribution. The analytical solution compells an "instantaneous" spread of temperature rise instantaneously at even far position x, that is, in no time. "In no time," so that the heat transfer has infinite speed much faster than light speed! Does this really happen? Can this really happen?

The starting base is the diffusion equation. The book does not take into account the microscopic interpretation of thermal phenomenon. The heat transfer may be interpreted microscopically as the statistical propergation of excess kinetic energy from one side to the lower side via the atomic contacts. Therefore I suppose that the heat tranfer speed cannot exceed the sonic speed of the rod. Or is the author Tolstov actually right? Does the heat run faster than light speed in the straight rod which is thermally insulated laterally? Are the contact processes between the atoms quantum mechanical and faster than the usual sonic speed? What is the truth? Pleae someone tell me the truth and resolve this puzzle!

A stray sheep

April 30, 2012

All models are approximations of reality. The heat equation assumes a continuum, ignoring the actual microscopic mechanisms which transfer energy. On a large enough scale, the continuum assumption is fairly good. However, as you have noted, there are situations where the assumptions break down. When you get down to the atomic-scale contacts, the continuum assumption is no longer valid. We need to modify the model to account for reality at this scale, or find a different model to use.

It may be useful to think of models and equations as descriptions of reality. A useful model gives us insight into the underlying rules, but one should not confuse the model for reality itself! The power of science lies in our willingness to test our models against reality through experiments and let the data guide us to an improved understanding.