152 Chapter 5. Life in the slow lane: the low Reynolds-number world[[Student version, December 8, 2002]]
car is initially rolling backwards, then hits a wall behind you and stops. Once again your head’s
acceleration pointsforward,asits velocity jumps from negative to zero. Once again your headrest
pushesforwardon your head. In other words,
In Newtonian physics the time-reversed process is a solution to the equations
of motion with the same sign of force as the original motion.(5.12)
In contrast, the viscous friction rule isnottime-reversal invariant: The time-reversed trajectory
doesn’t solve the equation of motion with the same sign of the force. Certainly a pebble in molasses
never fallsupward,regardless what starting velocity we choose! Instead, to get the time-reversed
motion we must apply a force that is time reversed andoppositein direction to the original. To
see this in the mathematics, let’s reconsider the equation of motion we found for diffusion with
drift,vdrift=f/ζ(Equation 4.12), and rephrase it using ̄x(t), the position of the particle at timet
averaged over many collision times. ( ̄x(t)shows us the net drift but not the much faster thermal
jiggling motion.) In this language our equation of motion reads
d ̄x
dt=
f(t)
ζ. (5.13)
The solution ̄x(t)toEquation 5.13 could be uniform motion (if the forcef(t)isconstant), or
accelerated motion (otherwise). But think about the time-reversed motion, ̄xr(t)≡x ̄(−t). We can
find its time derivative using the chain rule from calculus; it won’t be a solution of Equation 5.13
unless we replacef(t)by−f(−t).
The failure of time-reversal invariance is simply a signal that somethingirreversibleis happening
in frictional motion. Phrased this way, the conclusion is not surprising: We already knew that
friction is the one-way dissipation, or degradation, of ordered motion into disordered motion. Our
simple model for friction in Section 4.1.4 explicitly introduced this idea, via the assumption of
randomizing collisions.
Here is another example of the same analysis. Section 4.6 gave some solutions to the diffusion
equation (Equation 4.19 on page 118). Taking any solutionc 1 (x, t), we can consider its time-reversed
versionc 2 (x, t)≡c 1 (x,−t), or its space-reflected versionc 3 (x, t)≡c 1 (−x, t). Take a moment to
visualizec 2 andc 3 for the example shown in Figure 4.12a.
Your Turn 5e
Substitutec 2 andc 3 into the diffusion equation and see whether they also are solutions. [Hint:
Use the chain rule to express derivatives ofc 2 orc 3 in terms of those ofc 1 .] Then explain in
words why the answer you got is right.The distinction between fluids and solids also hinges upon their time-reversal behavior. Suppose
weput an elastic solid, like rubber, between the plates in Figure 5.2b. Then the force resisting
deformation follows aHooke relation,f=−k(∆z). The spring constantkin this relation depends
on the geometry of the sample, but for many materials we can write it ask=GA/d,where the
shear modulusGis a property only of the material.^4 Thus
f
A=−
(
∆z
d)
G. (5.14)
The quantityf/Ais called theshear stress,while (∆z)/dis theshear strain.Afluid, in contrast,
hasf/A=ηv/d(Equation 5.4).
(^4) Compare Equation 5.4, which was organized so thatηis also a property of the material, not the sample geometry.