THE REALITY OF MOLECULES 97
In this, Einstein's fundamental equation for Brownian motion, (x^2 ), t, a, 77 are
measurable; therefore TV can be determined. As mentioned earlier, one never
ceases to experience surprise at this result, which seems, as it were, to come out
of nowhere: prepare a set of small spheres which are nevertheless huge compared
with simple molecules, use a stopwatch and a microscope, and find Avogadro's
number.
As Einstein emphasized, it is not necessary to assume that all particles are at
the origin at t = 0. That is to say, since the particles are assumed to move inde-
pendently, one can consider n(x,t)dx to mean the number of particles displaced
by an amount between x and x + dx in t seconds. He gave an example: for water
at 17°C, a « 0.001 mm, N « 6 X 1023 , one has (x^2 )1/2 « 6 urn if t = 1
minute.
Equation 5.18 is the first instance of a fluctuation-dissipation relation: a mean
square fluctuation is connected with a dissipative mechanism phenomenologically
described by the viscosity parameter.
Einstein's paper immediately drew widespread attention. In September 1906
he received a letter from Wilhelm Conrad Roentgen asking him for a reprint of
the papers on relativity. In the same letter Roentgen also expressed great interest
in Einstein's work on Brownian motion, asked him for his opinion on Gouy's
ideas and added, 'It is probably difficult to establish harmony between [Brownian
motion] and the second law of thermodynamics' [R6]. It is hard to imagine that
Einstein would not have replied to such a distinguished colleague. Unfortunately,
Einstein's answer (if there was one) has not been located.
- Diffusion as a Markovian Process. All the main physics of the first Einstein
paper on Brownian motion is contained in Eq. 5.18. However, this same paper
contains another novelty, again simple, again profound, having to do with the
interpretation of Eq. 5.15. This equation dates from the nineteenth century and
was derived and applied in the context of continuum theories. In 1905 Einstein,
motivated by his reflections on Brownian motion, gave a new derivation of the
diffusion equation.
As was already done in the derivation of Eq. 5.12, assume (Einstein said) that
the suspended particles move independently of each other. Assume further that we
can define a time interval T that is small compared with the time interval of obser-
vation (t in Eq. 5.18) while at the same time T is so large that the motion of a
particle during one interval r does not depend on its history prior to the com-
mencement of that interval. Let $(A)<iA be the probability that a particle is dis-
placed, in an interval r, by an amount between A and A + c/A. The probability
0 is normalized and symmetric:
