9780192806727.pdf

STATISTICAL PHYSICS

Since the particles move independently, we can relate n(x,t + r)dx to the distri-

bution at time t by

Develop the Ihs to first order in T, the rhs to second order in A, and use Eq. 5.19.

Then we recover Eq. 5.15, where D is now defined as the second moment of the

probability distribution 0:

All information on the dynamics of collision is contained in the explicit form of

$(A). The great virtue of Eq. 5.18 is therefore that it is independent of all details

of the collision phenomena except for the very general conditions that went into

the derivation of Eq. 5.21.

Today we would say that, in 1905, Einstein treated diffusion as a Markovian

process (so named after Andrei Andreievich Markov, who introduced the so-called

Markov chains in 1906), thereby establishing a link between the random walk of

a single particle and the diffusion of many particles.

5. The Later Papers. I give next a brief review of the main points contained
   in Einstein's later papers on Brownian motion.

1. December 1905 [E8]. Having been informed by colleagues that the consid-
   erations of the preceding paper indeed fit, as to order of magnitude, with the
   experimental knowledge on Brownian motion, Einstein entitles his new paper 'On
   the Theory of Brownian Motion.' He gives two new applications of his earlier
   ideas: the vertical distribution of a suspension under the influence of gravitation
   and the Brownian rotational motion for the case of a rotating solid sphere. Cor-
   respondingly, he finds two new equations from which N can be determined. He
   also notes that Eq. 5.18 cannot hold for small values of t since that equation
   implies that the mean velocity, (x^2 )^^2 /t, becomes infinite as t — 0. 'The reason
   for this is that we ... implicitly assumed that, during the time t, the phenomenon
   is independent of [what happened] in earlier times. This assumption applies less
   well as t gets smaller.'


2. December 1906 [E9]. A brief discussion of 'a phenomenon in the domain
   of electricity which is akin to Brownian motion': the (temperature-dependent)
   mean square fluctuations in the potential between condensor plates.


3. January 1907 [Ell]. Einstein raises and answers the following question.
   Since the suspension is assumed to obey van 't HofFs law, it follows from the
   equipartition theorem that (v^2 ), the mean square of the instantaneous particle
   velocity, equals 3RT/mN (m is the mass of the suspended particle). Thus, (v^2 )

*The general solution for all ( was given independently by Ornstein [O2] and Fiirth [F3].

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