Here’s a reminder of where we’ve been recently and where we’re going next.
We’ve been touring through thermodynamics and statistical mechanics with Herb Callen and Harry Robertson, and in my previous post also with Mark Whittle, discussing bounded regions of space containing macroscopically homogeneous systems of particles.
More specifically, as Robertson says, we’ve looked at "the simple systems that are treated in terms of statistical thermophysics [and these] are made up of atoms or molecules, the spatial distribution of which is described by probability densities that are constant (or periodic on the scale of a crystal lattice) over the volume of the system."
Next, we’re going to be dealing with the probability densities of quantum mechanics and comparing the Callen and Robertson thermodynamics postulates with quantum mechanics postulates.
As a prelude to doing that, we’re going to now look at Dudley Towne’s discussion of “Waves confined to a limited region,” which is the title of Chapter 15 in his book Wave Phenomena, published in 1967, the year in which A Serious Man takes place. (Our beleaguered Professor Gopnick might have had a copy of it. He definitely has a copy of Dirac’s Principles of Quantum Mechanics on his desk when he hangs up on Sy Ableman after finding the failing student’s bribe money in the envelope.)
Specifically, we’re going from discussing atoms and molecules in some kind of container to discussing waves in some kind of container, where “container” means a bounded region of space. There are three parts to the discussion (the latter two parts will be coming along soon).
Part One: one-dimensional waves in a "box"
First we’ll consider waves on a string of length l, the classic classical-mechanical one-dimensional wave problem. Two dimensions of space, y and x, are needed to describe the motion, but it’s nevertheless called one-dimensional motion because the string only moves up and down (a particle on the string can move only in the ±y direction). Towne says that for the one-dimensional wave equation (with c as wave speed),
∂ 2y/∂x2 — (1/c2) ∂2y/∂t2 = 0
and boundary conditions y(0,t) = y(l,t) = 0, “an
important property of the general solution to this problem is that any function
satisfying all three of these conditions is necessarily periodic in time.”
For most of us, or at least for me, this seems to be the same old statement that sinusoidal waves with certain frequencies, the standing wave or resonant frequencies, satisfy these three conditions.
Aye, but Professor Towne is
saying more than that!
Sine waves on a string or in a box are not just periodic in time,
they’re also periodic in space. Towne’s statement only says “necessarily
periodic in time,” and he immediately gives a non-sinusoidal example of such a
confined wave:
Consider a short pulse which starts out near the center of the string and travels in the positive x-direction. This pulse is reflected with inversion at the fixed end x = l and is reinverted upon reflection at x = 0. When the pulse returns to the original position, traveling in the same direction that it was initially, the string is in identically the same condition as it was initially, and the process repeats itself. The period T is therefore the length of time required for the pulse to travel twice the length of the string: T = 2l/c. If the disturbance on the string is the superposition of any number of pulses traveling in either direction, they all repeat themselves in this interval and the entire motion is periodic.
Towne’s discussion should clear up any confusion you or I might
have had about the meaning of periodic boundary conditions. The waves that meet
the above three conditions may be periodic in space, but they don’t have to be.
They do have to be periodic in time, and this is one example of where the name “periodic
boundary conditions” comes from.*
That’s really all I want to point out here in Part One. But the
example of the pulse on a string is worth thinking about in terms of its
analogy to the quantum mechanical particle in a box, and its relation to a
classical or quantum harmonic oscillator, and the fact that such a pulse can be
expressed as a Fourier series (yeah, the pulse needs a specific profile in
order to do that). Also: do the coefficients in a Fourier series
expansion have any relation to the components of a
Hilbert space vector?
Postscript 30 October 2022
Here’s
an unexpected example of particles confined to a limited region that uses
periodic boundary conditions in one dimension. I learned about this example
from page 273 of the recently published textbook by Robert Swendsen, An Introduction
to Statistical Mechanics and Thermodynamics (Second Edition, 2020). We’ll be
coming back to Swendsen later, because he’s a former student of Herbert Callen
and gives a new version of Callen’s postulates of thermodynamics, with four “essential”
postulates and three “optional” postulates. The optional ones aren’t applicable
to all thermodynamic systems but simplify the analysis when they are applicable,
Swendsen says.
The
free expansion of a classical gas is the example of particles confined to a
limited region that so unexpectedly uses periodic boundary conditions in one dimension (well, maybe it meets your expectations, but I hadn't expected it).
It’s in Swendsen’s chapter on irreversibility, Chapter 22, and comes from a 1958
Physical Review paper by Harry Lloyd Frisch: “An approach
to equilibrium,” Phys. Rev., 109, 22-9 (1958). “Following Frisch’s 1958
paper,” Swendsen says, “we will eliminate the difficulties involved in
describing collisions with the walls at x= 0 and x= L by mapping
the problem onto a box of length 2L with periodic boundary conditions.” Yeah, I guess you can't call it a free expansion when the particles collide with walls. More about that later!
Post-postscript 25 November 2022
I wasn't aware until now that "periodic boundary conditions" of a more general nature are used in simulations and computer games. Now I know. Swendsen's discussion of Frisch's use of periodic boundary conditions for the free expansion of of an ideal gas is a simple example of a simulation.
