On Einstein's Doctoral Thesis

arXiv:physics/0504201v1 [physics.hist-ph] 27 Apr 2005

On Einstein’s Doctoral Thesis

∗

Norbert Straumann

Institute fo r Theoretical Physics University of Zurich,

CH–80 57 Zurich, Switzer land

February 2, 2008

Abstract

Einstein’s thesis “A New Determination of Molecular Dimensions”

was the second of his ﬁve celebrated papers in 1905. Although it is –

thanks to its widespread practical applications – the most quoted of

his papers, it is less known than the other four. The main aim of the

talk is to show what exactly Einstein did in his dissertation. As an

important application of the theoretical results for the viscosity and

diﬀusion of solutions, he got (after eliminating a calculational error) an

excellent value for the Avogadro number from data for sugar dissolved

in water. This was in agreement with the value he and Planck had

obtained from the black-body radiation. Two weeks after he ﬁnished

the ‘Doktorarbeit’, Einstein submitted his paper on Brownian motion,

in which the diﬀusion formula of his thesis plays a cru cial role.

1 Introduction

When Einstein’s great papers of 1905 appeared in print, he was not a new-

comer in the Annalen der Physik, where he published most of his early work.

Of crucial importance for his further research were three papers on the foun-

dations of statistical mechanics, in which he tried to ﬁll what he considered

to be a gap in the mechanical foundations of thermodynamics. At the time

when Einstein wrote his three papers he was not familiar with the work of

Gibbs and only partially with that of Boltzmann. Einstein’s papers form

a bridge, parallel to the Elementary Principles of Statistical Mechanics by

∗

Talk given at the joint colloquium of ETH and the University of Z¨urich, 27 April

(2005).

Gibbs in 190 2, between Boltzmann’s work and the modern approach to sta-

tistical mechanics. In particular, Einstein independently formulated the dis-

tinction between the microcanonical and canonical ensembles and derived the

equilibrium distribution for the canonical ensemble from the microcanonical

distribution. Of special importance for his later research was the derivation

of the energy ﬂuctuation formula for the canonical ensemble.

Einstein’s prof ound insight into the nature and size of ﬂuctuations played

a decisive role for his most revolutionary contribution to physics: the light-

quantum hypothesis. In this ﬁrst paper of 1905 he extracted the light-

quantum postulate from a statistical mechanical analogy between radiation

in the Wien regime and a classical ideal gas of material particles. In this con-

sideration Boltzmann’s principle, relating entropy and probability of macro-

scopic states, played a key role. Lat er Einstein extended these considerations

to an analysis of ﬂuctuations in the energy and momentum of the radia-

tion ﬁeld. Fo r the latter he was also drawing on ideas and methods he had

developed in the course of his work on Brownian motion, another beauti-

ful application of ﬂuctuation theory. This deﬁnitely established the reality

of atoms and molecules, and, more generally, gave strong support for the

molecular-kinetic theory of thermodynamics.

Einstein’s Doctoral thesis “A New Determination of Molecular Dimen-

sions” was the second o f his celebrated ﬁve papers in 1905. Unfortunately, it

is not suﬃciently well known. The main body of the paper is devoted to the

hydrodynamic derivation of a relation between the coeﬃcients of viscosity

of a liquid with and without susp ended particles. In addition, Einstein de-

rived a novel formula for the diﬀusion constant D of suspended microscopic

particles. This was o bta ined on the basis of thermal and dynamical equi-

librium conditions, making use of van’t Hoﬀ’s law for the osmotic pressure

and Stokes’ law for the mobility of a particle. Einstein then applied these

two relations to sugar dissolved in water. Using empirical data he got (after

eliminating a calculational error) an excellent value of the Avogadro number

and an estimate of the size o f sugar molecules. Einstein’s thesis is the most

quoted among his papers.

Soon afterwards Einstein’s diﬀusion formula became also important in his

work on Brownian motion. In t his celebrated paper he ﬁrst gave a statistical

mechanical foundation of the o smotic pressure, and then repeated his earlier

derivation in the thesis.

2 Biographical re marks

Einstein devoted his thesis to his friend Marcel Grossmann. Before I come

to a technical discussion of the paper, I would like to give some biographical

and other background.

Until 1909 the ETH was not authorized to grant doctoral degrees. For

this reason a special arr angement enabled ETH students to obtain doctorates

from the University. At the time most dissertations in physics by ETH

students were carried out under the supervision of H.F. Weber, Einstein’s

former teacher at the ‘Polytechnikum’ a s it wa s then called. The University

of Z¨urich had only one physics chair, held by Alfred Kleiner. His main

research was focused on measuring instruments, but he had an interest in

the foundations of physics. From letters to Mileva one can see that Einstein

often had discussions with Kleiner on a wide range of topics. Einstein also

showed him his ﬁrst dissertation in November 1901. This dissertation has not

survived, and it is not really clear what it contained. At any rate, Einstein

withdrew his dissertation in February 1902. O ne year later he was giving up

his plan to obta in a doctorate. To Besso he wrote: “the whole comedy has

become tiresome for me”.

By March 1903 he seems to have changed his mind. Indeed, a letter to

Besso contains some of the central ideas of the 1905 dissertation, especially

in the second part of the following quote:

“Have you a l read y calculated the absol ute magnitude of ions on

the assumption that they are spheres and so large that the hy-

drodynamical equations for vis cous ﬂuids are a pplicable? With

our knowledge o f the absolute magnitude of the electron [ c harge]

this would be a simple matter ind eed. I would have done it my-

self but lack the reference material and the time; you could also

bring in diﬀusion i n order to obtain information about neutral

salt molecules in solution.”

Kleiner was, of course, one of the two faculty r eviewers of the dissertation,

submitted by Einstein to the University on 20 July, 19 05. His judgement

was very positive: “the arguments and calculations to be carried out are

among the most diﬃcult in hydrodynamics”. The other reviewer, Heinrich

Burkhardt, Professor f or Mathematics at the University, added: “the mode

of treatment demonstrates fundamental mastery of the relevant mathematical

methods.”

In his biogra phy of Einstein, Carl Seelig repor ts: “Einstein later laugh-

ingly recounted that his dissertation was ﬁrst returned by Kleiner with the

comment that it was too short. After he had added a single sentence, it wa s

accepted without further comment.”

The physical reality of atoms was not yet universally accepted by the

end of the nineteenth century. Fervent opponents were Wilhelm Ostwald

and Georg Helm (who called themselves “energeticists”), and Ernst Mach

admitted only that atomism may have a heuristic or didactic utility.

In his ﬁrst three papers of 1905, Einstein found three diﬀerent methods

of determining the Avogadro number. (A few years later he found another

one in his study of critical opalescence.) For him this was not only important

for establishing t he existence of atoms. He later wrote to Perrin:

“A precis e determination o f the si ze of molecules seems to me of

the highest importance because Planck’s rad i ation formula can be

tested more precisely through such a determina tion than through

measurements on radi ation.”

3 Einstein’ s di ssertation

By 1905 several methods for determining molecular sizes were developed.

The most reliable ones were based on kinetic t heory of g ases. An important

early example is Loschmidt’s work from 1 865.

The following introductory

remarks in Einstein’s dissertation indicate what he adds to this.

“The earliest determinations of real sizes of molecules w ere poss i -

ble by the kinetic theory of gases, whereas the physical phenomena

observed in liquids have thus far not served for the d etermination

of m olecular sizes. This is no doubt due to the fact that it has

not yet been possible to overcome the obstacles that impede the

development of a detailed molecular-kinetic theory of liquids. It

will be shown i n this paper that the size of molecules of substances

dissolved in an undissociated dilute solution can be obtained f rom

the internal friction of the solution and the pure solvent, and from

the d i ﬀusion of the dissolved substance within the solvent. (...).”

Beside originality and intuition, great scientists usually also dispose of a

fair amount of technical abilities. That Einstein was not an exception in this

respect, should become clear if we now go into the technical details of his

dissertation.

J. Loschmidt, Wiener Ber. 52, 395 (1866). See also J.C. Maxwell, Collected Works,

Vol. 2, p. 361.

3.1 Basic equations of hydrodynamics

Let me ﬁrst recall some general facts of hydrodynamics, that we shall need.

I will use nota tio n that has become standard, and not the one that was

common at the time when Einstein did his work.

For stationary incompressible ﬂows of homogeneous ﬂuids, the Navier-

Stokes equation is

(v · ∇)v = −

∇p +

△v.

We consider only situations with small Reynold numbers.Then one can ne-

glect the left-hand side, and t he basic equations become

∇p = −η△v, ∇ · v = 0. (1)

These imply that t he pressure is harmonic: △p = 0. The same holds for the

vorticity curl v. We also recall the expression for the stress tensor

= −pδ

+ η(∂

+ ∂

). (2)

According to (1) this is divergence-free: ∂

= 0. Later we shall also need

the following expression for the rate W at which the stresses do work on the

surface ∂Ω bounding a region Ω:

W =

∂Ω

dA. (3)

Here, n is the outward pointing unit vector.

3.2 Einstein’s strategy

With Einstein we now consider an incompressible ﬂuid of viscosity η

, in

which a large number of identical, rigid, spherical part icles is inserted. This

suspension can be described in two ways: (1) On large scales, in comparison

to the average separation of neighboring solute part icles, as a homogeneous

medium with an eﬀective viscosity η. (2) By the stationary ﬂow of the ﬂuid

(solvent) that is modiﬁed by the suspended particles.

For both descriptions Einstein computes according to (3) the rate of work

for a big region Ω and obtains by equating the two results t he important

formula

η = η



1 +



, (4)

where ϕ denotes the fraction of the volume occupied by the suspended parti-

cles. This is assumed t o be small (dilute suspension). (D ue to a calculational

error, Einstein originally lost the factor 5/2; we shall come back to this amus-

ing story.)

3.3 Velocity ﬁeld for a single suspended particle

We ﬁrst adopt the second description. As a preparing task we have to deter-

mine t he modiﬁcation of a ﬂow with constant velocity gradient, say, caused

by a single little ball. Mathematically, we have to solve a boundary value

problem for the elliptic system (1).

So let the unperturbed velocity ﬁeld be

(0)

= e

, (5)

where e

is a constant, symmetric, traceless tensor. The last property reﬂects

the incompressibility. e

is the deformation t ensor (we are not interested in

ﬂows with non-vanishing vorticity). The unperturbed pressure is denoted by

(0)

. The stress tensor for the background ﬂow is

(0)

= −p

(0)

+ 2η

. (6)

We decompose the modiﬁed velocity ﬁeld v according to

v = v

(0)

+ v

(1)

(7)

into an unperturbed part plus a perturbation v

(1)

. The boundary conditions

are: v = 0 on the ball with radius a and v = v

(0)

at inﬁnity. Analogous

decompositions are used for the pressure and the stresses:

p = p

(0)

+ p

(1)

, σ

= σ

(0)

+ σ

(1)

, (8)

where

(1)

= −p

(1)

+ η

(∂

(1)

+ ∂

(1)

). (9)

For W we then have the decomposition (|Ω|= volume of Ω)

W = 2η

|Ω| + e

∂Ω

(1)

dA +

∂Ω

(1)

(0)

dA. (10)

The rigid ball is taken as the origin of a cartesian coordinate system.

Einstein determines the perturbations v

(1)

and p

(1)

with the help o f a metho d

which is described in Kirchhoﬀ’s “Vorlesungen ¨uber Mechanik”

, which he

had studied during his student years. This involves the following two steps:

a) Determine a function V , which satisﬁes the equation

△V =

(1)

, (11)

G. Kirchhoﬀ, Vorlesungen ¨uber mathematische Physik, Vol. 1, Mechanik, Teubner

(1897).

and set

(1)

= ∂

V + v

′

, (12)

where v

′

has to satisfy the following equations

△v

′

= 0, ∂

′

= −

(1)

. (13)

Remark on a). As a consequence of (11)-(13) the basic equations for v

(1)

and

(1)

are satisﬁed:

△v

(1)

= η

∂

△V = ∂

(1)

, ∂

(1)

= △V + ∂

′

= 0.

b) Use the following decaying harmonic ansatz for p

(1)

= Ae

∂





(14)

with a constants A, and try for v

′

the harmonic expression

′

= −

∂





+ B∂

∂





. (15)

This fulﬁlls both equations (13) for

A = A, because we then have

∂

′

= −Ae

∂





= −

(1)

As a result of △r = 2/r, equation (11) is satisﬁed for

V =

∂

r. (16)

Performing the diﬀerentiations in (15) and (16 ) , we obtain

(1)

+ B



− 15e



. (17)

The boundary condition v

(1)

= −e

for r = a requires

A = −

, B = −

. (18)

We thus obtain for the perturbation of the velocity ﬁeld (n

:= x

/r)

(1)

= −

−



− 15e



. (19)

According to (12) and (15) for

A = A we can represent v

(1)

also as follows

(1)

= −

∂

(r) +

∂





−

∂





. (20)

Equation (14) gives fo r the pressure

p = p

(0)

− 5η

. (21)

Einstein claims that it can be demonstrated that his solution of the

boundary value problem is unique, but he gives only some indications of

what he thinks is a proof. Apparently, he did not know that an elegant

uniqueness proof for such problems was already given in 1868 by Helmholtz.

Consider for two solutions of the basic equations, for given velocity ﬁelds on

the boundaries, the non-negative quant ity (θ

′

− θ

)(θ

′

− θ

), where θ

′

, θ

are the deformation tensors of the two velocity ﬁelds. It is easy to show

that the integral of this function over the region outside the bodies must

vanish. (Use partial integrations and the basic equations (1).) Therefore,

′

= θ

. In other words, the deformation tensor for the diﬀerence v

′

− v

of the two velocity ﬁelds vanishes. This diﬀerence is thus a combination of

a rigid t r anslation and a rigid rotation. Because of the imposed boundary

conditions, the two velocity ﬁelds must agree. The pressures for the two

solutions are, t herefore, also the same, up to an additive constant.

3.4 Two expressions for the rate W

In (10) we now choose for Ω a large ball K

with radius R. In leading order

only the ﬁrst term of (19) contributes, and a routine calculation leads to the

following expression for W in terms of the spherical moments

4π

dΩ =

(δ

+ δ

W = 2η

|Ω| + 20πa

{3e

− e

}

= 2η



|Ω| +

4π



This holds for a single ball. As long as the suspension is dilute, we obtain

Einstein’s (corrected) result

W = 2η

|Ω|



1 +



. (22)

H. Helmholtz, “Theorie der station¨aren Str¨ome in reibenden Fl ¨ussigkeiten”, Wiss.

Abh., Bd. I, S. 223.

Following Einstein, we now calculate the same quant ity by adopting t he

ﬁrst description of the suspension. For this we write the result (20) in the

form

= e

+ (e

△ − e

∂

)∂

f, (23)

with

f = −

Ar −

(24)

(A and B have the earlier meaning (18)). When the contributions of all t he

suspended balls, with number density n, inside the ball K

are summed, we

obtain for the velocity ﬁeld in K

= e

+ (e

△ − e

∂

)∂

F, (25)

where

F (|x|) = n

f(|x − x

′

|) d

′



− r



− 2πnB



−



(26)

From this one easily ﬁnds

= e

(1 − ϕ). (27)

Einstein obtained this result slightly diﬀerently. We can use it to obtain a

second expression for W :

W = 2ηe

|Ω|(1 − 2ϕ). (28)

Comparing (22 ) and (28) leads to the announced formula (4) due to Einstein.

3.5 Two relations between the Avogadro number

and the molecular radius.

If the rigid balls are molecules, for instance sugar, then

ϕ =

4π

, (29)

where ρ

is t he mass density of the solute and m

its molecular weight which

were known to Einstein. In addition, there existed measurements o f η/η

for

dilute sugar solutions. Hence, Einstein obtained from (4) a relation between

and a.

Using available data, Einstein states the following.

“One gram of sugar

dissolved in water has the same eﬀect on the coeﬃcient of viscosity as do small

I give here the later numbers from 1911.

suspended rigid spheres of a total volume o f 0.98 cm

.” On the other hand,

the density of an aqueous sugar solution behaves experimentally as a mixture

of water and sugar in dissolved form with a speciﬁc volume of 0.61 cm

. (The

latter is also the volume of one gram of solid sugar.) Einstein interprets the

diﬀerence of the two numbers as due to an attachment of water molecules to

each sugar molecule. The radius a in (29) is thus a “hydrodynamically eﬀec-

tive radius” of the molecule, which takes the enlargement due to hydration

into account.

Diﬀusion

In order to be able to determine the two quantities individually, Einstein

searched f or a second connection, and thereby found his famous diﬀusion

formula. Its derivation is quite short, but “extremely ingenious” (A. Pais).

It rests o n thermal and mechanical equilibrium considerations.

Assume that a constant external force f acts on the the suspended parti-

cles. This causes a particle current of magnitude nv, where n is the number

density and v = the velocity of the particle current. In equilibrium this

is balanced by the diﬀusion current −D∇n, D = diﬀusion constant. The

velocity of the pa rt icle current is proportional to f ,

v = bf, b : mobility. (30)

These considerations give us the (dynamical) equilibrium condition

D∇n = nbf. (31)

In thermal equilibrium, the external force is balanced by the gradient of

the osmotic pressure. According to the law of van’t Hoﬀ

this means

f =

∇n. (32)

Inserting this into the last relation leads to the simple formula

D = kT b. (33)

For the mobility Einstein uses Stokes’ relation

b =

6πη

(34)

According to this, the osmotic pressur e p exerted by the suspended particles is e xactly

the same as if they alone were present as an ideal gas. In equilibrium we thus have

nf = ∇p = kT ∇n.

and obtains in this way his famous formula

D =

6πη

, k =

(35)

(R = gas constant).

This was almost simultaneously discovered in Australia by William Suther-

land.

A beauty of the argument is tha t the exterior force drops out. Similar

equilibrium considerations between systematic and ﬂuctuating forces were

repeatedly made by Einstein.

3.6 Silence, a calculational error, late attention

By 1909 Perrin’s careful measurements of Brownian motion led to a new value

for Avogadro’s number that was signiﬁcantly diﬀerent from the va lue Einstein

had obtained from his thesis work, and also somewhat diﬀerent from what

he and Planck had deduced from black-body radiation. Einstein then drew

Perrin’s attentio n to his hydrodynamical method, and suggested its applica-

tion to the suspensions studied by Perrin. Then Jacques Bancelin, a Pupil

of Jean Perrin, checked Einstein’s viscosity fo r mula η = η

(1 + ϕ). Bancelin

conﬁrmed that there was an increase of the viscosity that was independent

of the size of the suspended particles, and only depends on the total vo lume

they occupy. However, he got a stronger increase. Initially, this increase was

too steep; in the publication Bancelin gives the result η = η

(1 + 2.9ϕ).

On 27 December, 1 910 Einstein wrote from Z¨urich to his former student

and collaborator Ludwig Hopf about the puzzling situation, a nd then adds:

“I have checked my previ ous calculations and arguments and found

no error in them. You would be doing a great service in this mat-

ter if you would carefully recheck my investigation. Either there

is an error in the wo rk, or the volume of Perrin’s suspend ed sub-

stance in the suspended state is greater than Perrin believ es . ”

Hopf indeed found an error in some diﬀerentiation process, and got the

formula (4). Einstein communicated the result to Perrin, and published in

(1911) a correction of his thesis in the Annalen. (By the way, this correction

is the second most quoted paper of Einstein.) New experimental data for

sugar solutions now gave the excellent va lue

= 6.56 × 10

(36)

for the Avogadro number, in good agreement with the results of other meth-

ods, in particular with Perrin’s determination from the Brownian motion, for

which he got the Nobel price in 1926. Both results were discussed by Perrin

in his extensive report at the famous Solvay conference in 1911.

4 Final remarks

In his ‘Autobiographical Notes’ of 1949, what he called his ‘necrology’, Ein-

stein only brieﬂy describes his applications of classical statistical mechanics.

The thesis is not mentioned at all. About the law of Brownian motion he

says:

“The agree ments of these considerations wi th ex perience together

with Planck’s determination of the true mol ec ular size from the

law of radiation (for high temperatures) convinced the sceptics,

who were quite numerous at the time (Ostwald, Mach) of the re-

ality of atoms. The antipathy of these scholars toward atomic the-

ory can indubitably be traced back to their positivistic philos ophi-

cal attitude. This is an interesting example of the fact that e ven

scholars of audacious spirit and ﬁne instinct can be obstructed in

the interpretation of fac ts by philo sophical prejudices.”

Perrin’s famous book “Les Atomes” o f 1913, a classic of twentieth century

physics

, ends with the words:

“The atomic theory has triumphed. Until recently still numerous,

its adversaries, at last overcome, now renounce one after another

their misgivings, which were, for so long, both legitimate and un-

deniably useful.”

Einstein’s very decent value (36) is not quoted in Perrin’s book. This

indicates that Einstein’s thesis was not widely appreciated in the early years.

For this reason Einstein published in 1920 a brief note, drawing attention to

his erratum from 1911, “which till now seems to have escaped the attention

of all who work in this ﬁeld”.

Since Einstein was so fond of applying physics to practical situations, he

would certainly have enjoyed hearing that his doctoral thesis found so many

applications.

A new edition of the original text has appeared in Fla mmarion (1991), ISBN 2-08-

081225-4; fo r an English translation, see: J. Perrin, Atoms, Van Nostrand (1916).