108 Chapter 4. Random walks, friction, and diffusion[[Student version, December 8, 2002]]
Once we measureζexperimentally, Equation 4.13 then connects it to a microscopic quantity,
the collision time ∆t.Wecan then substitute this value back into Idea 4.5b to find the effective
step sizeL,using the measuredD.
Recovering the familiar friction law (Equation 4.12) strengthens the idea that friction originates
in randomizing collisions of a body with the thermally disorganized surrounding fluid. Our result
goes well beyond the motion of Robert Brown’s pollen grains:Anymacromolecule, small dissolved
solute molecule, or even the molecules of water itself are subject to Equations 4.6 and 4.12. Each
typeof particle, in each type of solvent, has its own characteristic values ofDandζ.
Unfortunately, however, our theory hasnotmade a falsifiable, quantitative prediction yet. It
lets us compute the microscopic parametersLand ∆tof the random walk’s steps, but these are
unobservable! To test of the idea that diffusion and friction are merely two faces of thermal motion,
wemust take one further step.
Einstein noticed that there’s athirdrelation involving the two unknown microscopic parameters
Land ∆t.Toget it, we simply note that (L/∆t)^2 =(v 0 )^2. Our discussion of the ideal gas law
(Idea 3.21 and the discussion preceding it) concluded that the average value of this quantity is just
kBT/m.Combined with Equations 4.5b and 4.13, this relationoverdeterminesLand ∆t:The three
relations in these two unknowns can only hold ifDandζthemselves satisfy a particular relation.
This relation between experimentally measurable quantities is the falsifiable prediction we were
seeking. To find it, consider the productζD.
Your Turn 4c
Put all the pieces together: Continuing to work in one dimension, use Equations 4.5b and 4.13
to expressζDin terms ofm, L,and ∆t. Then use the definitionv 0 =L/∆t,and Idea 3.21 on
page 74, to show that
ζD=kBT. Einstein relation (4.15)Equation 4.15 is Einstein’s 1905 result.^5 It states that thefluctuationsin a particle’s position are
linked to thedissipation,or frictional drag, it feels. The connection is quantitative anduniversal:
it’s always given by the same quantitykBTappearing in the ideal gas law, no matter what sort
of particle we study. For example, the right-hand side of Equation 4.15 does not depend on the
massmof the particle. Smaller particles will feel less drag (smallerζ)but will diffuse more readily
(biggerD)insuch a way that all particles obey Equation 4.15.
Previously we had regardedDandζas two independent quantities, obtained empirically from
twovery different sorts of experiments. The Einstein relation says that on the contrary, they
are strictly related to each other, a testable prediction of the hypothesis that heat is disordered
molecular motion. For example, whereas bothζandDcan have extremely complicated dependence
on the temperature, Equation 4.15 says theirproductdepends onTin a very simple way. We can
check whether various kinds of particles, of various sizes, at various temperatures, all give exactly
the same value ofkB.They do; you’ll see one example in Problem 4.5.
Einstein also checked whether the experiment he was proposing wasactually doable.He reasoned
that in order to see a measurable displacement of a single 1μmcolloidal particle, we’d have to wait
until it had moved several micrometers. If the waiting time for such a motion were impracticably
(^5) This relation was also derived and published in the same year by W. Sutherland, and obtained independently at
around the same time by M. Smoluchowski. Smoluchowski waited to do the experiments first before publishing the
theory, and so got scooped by Einstein and Sutherland.