A note added after the lecture
“I should like to add something that I didn’t have time for in the
lecture. (I always seem to prepare more than I have time to tell about.)
As I mentioned earlier, I got interested in a problem while working on
this lecture. I want to tell you what that problem is. Among the minimum
principles that I could mention, I noticed that most of them sprang in
one way or another from the least action principle of mechanics and
electrodynamics. But there is also a class that does not. As an example,
if currents are made to go through a piece of material obeying
Ohm’s law, the currents distribute
themselves inside the piece so that the rate at which heat is generated
is as little as possible. Also we can say (if things are kept
isothermal) that the rate at which energy is generated is a minimum.
Now, this principle also holds, according to classical theory, in
determining even the distribution of velocities of the electrons inside
a metal which is carrying a current. The distribution of velocities is
not exactly the equilibrium distribution [Chapter 40,
Vol. I, Eq. (40.6)] because they are drifting sideways. The
new distribution can be found from the principle that it is the
distribution for a given current for which the entropy developed per
second by collisions is as small as possible. The true description of
the electrons’ behavior ought to be by quantum mechanics, however. The
question is: Does the same principle of minimum entropy generation also
hold when the situation is described quantum-mechanically? I haven’t
found out yet.
 “The question is interesting academically, of course. Such principles
are fascinating, and it is always worthwhile to try to see how general
they are. But also from a more practical point of view, I want to
know. I, with some colleagues, have published a paper in which we
calculated by quantum mechanics approximately the electrical resistance
felt by an electron moving through an ionic crystal like NaCl.
[Feynman, Hellwarth, Iddings,
and Platzman, “Mobility of Slow Electrons in a Polar Crystal,”
Phys. Rev. 127, 1004 (1962).] But if a minimum
principle existed, we could use it to make the results much more
accurate, just as the minimum principle for the capacity of a condenser
permitted us to get such accuracy for that capacity even though we had
only a rough knowledge of the electric field.”
