Einstein’s
Miraculous
Argument of 1905
John D. Norton
Department of History and Philosophy of Science
Center for Philosophy of Science
University of Pittsburgh
1
Win $$ from your friends…
Q.
For which work,
published in 1905,
was Albert Einstein
awarded his
doctoral degree?
2
Warm Up
The Papers of Einstein’s
Year of Miracles, 1905
3
The Papers of 1905
1
"Light quantum/photoelectric effect paper"
2
Einstein's doctoral dissertation
3
"Brownian motion paper."
4
Special relativity
5
E=mc2
"On a heuristic viewpoint concerning the production and
transformation of light."
Annalen der Physik, 17(1905), pp. 132-148.(17 March 1905)
"A New Determination of Molecular Dimensions"
Buchdruckerei K. J. Wyss, Bern, 1905. (30 April 1905)
Also: Annalen der Physik, 19(1906), pp. 289-305.
"On the motion of small particles suspended in liquids at
rest required by the molecular-kinetic theory of heat."
Einstein inferred from the thermal properties of high
frequency heat radiation that it behaves
thermodynamically as if constituted of spatially localized,
independent quanta of energy.
Einstein used known physical properties of sugar
solution (viscosity, diffusion) to determine the size of
sugar molecules.
Einstein predicted that the thermal energy of small
particles would manifest as a jiggling motion, visible
under the microscope.
Annalen der Physik, 17(1905), pp. 549-560.(May 1905; received 11 May 1905)
“On the Electrodynamics of Moving Bodies,”
Annalen der Physik, 17 (1905), pp. 891-921. (June 1905; received 30 June, 1905)
“Does the Inertia of a Body Depend upon its Energy
Content?”
Annalen der Physik, 18(1905), pp. 639-641. (September 1905; received 27
September, 1905)
Maintaining the principle of relativity in
electrodynamics requires a new theory of space and
time.
Changing the energy of a body changes its inertia in
accord with E=mc2.
Einstein to Conrad Habicht
18th or 25th May 1905
“…and is very revolutionary”
Einstein’s assessment of his light quantum paper.
…So, what are you up to, you frozen whale, you smoked, dried, canned piece of sole…?
…Why have you still not sent me your dissertation? …Don't you know that I am one of the
1.5 fellows who would read it with interest and pleasure, you wretched man? I promise you
four papers in return…
The [first] paper deals with radiation and the energy properties of light
and is very revolutionary, as you will see if you send me your work first.
The second paper is a determination of the true sizes of atoms from the diffusion and the
viscosity of dilute solutions of neutral substances.
The third proves that, on the assumption of the molecular kinetic theory of heat, bodies on
the order of magnitude 1/1000 mm, suspended in liquids, must already perform an
observable random motion that is produced by thermal motion;…
The fourth paper is only a rough draft at this point, and is an electrodynamics of moving
bodies which employs a modification of the theory of space and time; the purely
kinematical part of this paper will surely interest you.
Why is only the light quantum
“very revolutionary”?
2
Einstein's doctoral dissertation
3
"Brownian motion paper."
4
Special relativity
5
E=mc2
"A New Determination of Molecular Dimensions"
Buchdruckerei K. J. Wyss, Bern, 1905. (30 April 1905)
Also: Annalen der Physik, 19(1906), pp. 289-305.
All the rest develop or
complete 19th century physics.
Advances the molecular kinetic
program of Maxwell and
Boltzmann.
"On the motion of small particles suspended in liquids at
rest required by the molecular-kinetic theory of heat."
Annalen der Physik, 17(1905), pp. 549-560.(May 1905; received 11 May 1905)
“On the Electrodynamics of Moving Bodies,”
Annalen der Physik, 17 (1905), pp. 891-921. (June 1905; received 30 June, 1905)
“Does the Inertia of a Body Depend upon its Energy
Content?”
Annalen der Physik, 18(1905), pp. 639-641. (September 1905; received 27
September, 1905)
Establishes the real significance
of the Lorentz covariance of
Maxwell’s electrodynamics.
Light energy has momentum;
extend to all forms of energy.
7
Why is only the light quantum
“very revolutionary”?
All the rest develop or
complete 19th century physics.
Well, not always.
"Monochromatic radiation of low density
behaves--as long as Wien's radiation
formula is valid [i.e. at high values of
frequency/temperature]--in a
thermodynamic sense, as if it consisted
of mutually independent energy quanta
of magnitude [h]."
The great achievements of 19th
century physics:
•The wave theory of light; Newton’s
corpuscular theory fails.
•Maxwell’s electrodynamic and its
development and perfection by
Hertz, Lorentz…
•The synthesis: light waves just are
electromagnetic waves.
Einstein’s light quantum paper
initiated a reappraisal of the physical
constitution of light that is not
resolved over 100 years later.
8
Goals
of this presentation
9
Historical-methodological…
The content of Einstein’s
discovery was quite unanticipated:
High frequency light energy exists in
• spatially independent,
• spatially localized
points.
The method of Einstein’s discovery
was familiar and secure.
Einstein’s research program in statistical
physics from first publication of 1901:
How can we infer the microscopic
properties of matter from its macroscopic
properties?
The statistical papers of 1905: the
analysis of thermal systems consisting of
• spatially independent
• spatially localized,
points.
(Dilute sugar solutions,
Small particles in suspension)
10
If….
If we locate Einstein’s light quantum
paper against the background of his work
in statistical physics,
its methods are an inspired variation of
ones repeated used and proven effective
in other contexts on very similar
problems.
If we locate Einstein’s light
quantum paper against the
background of electrodynamic
theory, its claims are so far
beyond bold as to be foolhardy.
11
Foundational…
Einstein’s visceral
mastery of
Why we need atoms
and not just ordinary
thermodynamics.
fluctuations.
The thermodynamics
of fluctuating, few
component systems.
How to compute
fluctuations.
“Boltzmann’s
Principle.”
12
Einstein’s Early Program
in Statistical Physics
13
Einstein’s first two “worthless” papers
Einstein to Stark, 7 Dec 1907, “…I am
sending you all my publications
excepting my two worthless beginner’s
works…”
“Conclusions drawn
from the
phenomenon of
Capillarity,”
Annalen der Physik,
4(1901), pp. 513523.
“On the thermodynamic
theory of the difference
in potentials between
metals and fully
dissociated solutions of
their salts and on an
electrical method for
investigating molecular
forces,” Annalen der
Physik, 8(1902), pp.
798-814.
14
Einstein’s first two “worthless” papers
Einstein’s hypothesis:
Forces between molecules at distance
r apart are governed by a potential P
satisfying
From
macroscopic properties of
capillarity and
electrochemical potentials
P = P - cc(r)
for constants cand c characteristic
of the two molecules and universal
function (r).
infer
coefficients in the
microscopic force law.
Equilibration of osmotic pressure by a field instead of a semi-permeable
membrane was a device Einstein used repeatedly but casually in 1905, but had
been introduced with great caution and ceremony in his 1902 “Potentials” paper.
15
Independent Discovery of the Gibbs Framework
3 papers 19021904
Einstein, Albert.
'Kinetische Theorie des
Waermegleichgewichtes und
des zweiten Hauptsatzes der
Thermodynamik'. Annalen
der Physik, 9 (1902)
16
Independent Discovery of the Gibbs Framework
3 papers 19021904
Einstein, Albert. 'Eine
Theorie der Grundlagen
der Thermodynamik'.
Annalen der Physik, 9
(1903)
17
Independent Discovery of the Gibbs Framework
3 papers 19021904
Einstein, Albert. 'Zur
allgemeinen molekularen
Theorie der Waerme'.
Annalen der Physik, 14
(1904)
18
The Hidden Gem
19
Einstein’s Fluctuation Formula
Any canonically distributed
system
 E 
p(E)  exp  
 kT 

Variance of energy from mean
dE
  (E  E)  kT
 kT 2C
dT
2
2
2
Heat capacity is
macroscopically
measureable.
(1904)
20
Applied to an Ideal Gas
Ideal monatomic gas, n molecules
3nkT
E
2
3nk
C
2
dE
  (E  E)  kT
 kT 2C
dT
2
2
2

rms deviation of energy from mean

n=1024…. negligible
3n / 2kT
1


E (3n / 2)kT
3n / 2
n=1
1
 0.816
3/ 2

21
In 1904, no one had any solid
idea of the constitution of
heat radiation!
Applied to heat radiation
Volume V of heat radiation (Stefan-Boltzmann law)
E  VT 4 C  4VT 3
 2  (E  E)2  kT 2
dE
 kT 2C
dT

rms deviation of energy from mean


2 kVT 5/2
k 1

2
4
E
VT
V T 3/2
Hence estimate volume V in which
fluctuations are of the size of the
mean energy.
 E
2
2
(1904)
22
Einstein’s Doctoral
Dissertation
"A New Determination of
Molecular Dimensions”
23
24
How big are molecules?
= How many fit into a gram mole? = Loschmidt’s number N
Find out by determining how the
presence of sugar molecules in dilute
solutions increases the viscosity of
water. The sugar obstructs the flow and
makes the water seem thicker.
After a long and very hard calculation…
And after very many special assumptions…
Apparent viscosity m*
=viscosity m of pure water
x (1 + fraction of volume
taken by sugar )
= (r/m) N (4p/3) P 3
r= sugar density in the solution
m = molecular weight of sugar
P = radius of sugar molecule, idealized as a sphere
Well, not quite. Einstein made a
mistake in the calculation. The
correct result is
m* = m (1 + 5/2 )
The examiners did not notice.
Einstein passed and was awarded the
25
PhD. He later corrected the mistake.
Recovering N
Turning the expression for apparent viscosity inside out:
N = (3m/ 4pr) x (m*/m - 1) x 1/P 3
All
measurable
quantities
P = radius
of
molecule.
Unknown!
ONE equation in TWO unknowns.
Einstein needs another equation.
The rate of diffusion of sugar in water is fixed
by the measurable diffusion coefficient D.
Einstein shows:
N = (RT/6pmD) x 1/P
All
measurable
quantities
TWO equations in TWO unknowns.
Einstein determined
N = 2.1 x 10 23
After later correction for his calculation error
N = 6.6 x 10 23
26
The statistical physics of dilute sugar solutions
Sugar in dilute solution consists of
a fixed, large number of component
molecules that do not interact with
each other.
Hence they can be treated by exactly
the same analysis as an ideal gas!
Sugar in dilute solution exerts an
osmotic pressure P that obeys the
ideal gas law
PV = nkT
Dilute sugar solution in a gravitational field.
27
Recovering the equation for diffusion
The equilibrium sugar concentration
gradient arises from a balance of:
Sugar molecules falling
under the effect of gravity.
Stokes’ law F = 6pmPv, F = gravitational force
And
Sugar molecules diffusing
upwards because of the concentration
gradient.
density
gradient
pressure
gradient
(ideal gas law)
upward
force
The condition for perfect balance is
N = (RT/6pmD) x 1/P
28
“Brownian motion paper.”
“On the motion of small particles suspended
in liquids at rest required by the molecularkinetic theory of heat.”
29
30
An easier way to estimate N?
Doctoral dissertation: Einstein determined N from TWO equations in TWO unknowns N, P.
N = (3m/ 4pr) x (m*/m - 1) x 1/P 3
N = (RT/6pmD) x 1/P
What if sugar molecules were so big that we could measure their diameter P under the microscope?
Why not just do the same analysis with microscopically visible particles?! Then P is observable.
Only ONE equation is needed.
Particles in suspension = a fixed, large number of
component that do not interact with each other.
Hence they can be treated by exactly the same
analysis as an ideal gas and dilute sugar solution!
The particles exert a pressure due to their thermal motions.
PV=nkT
…and this leads to their diffusion according to the
same relation N = (RT/6pm D) x 1/P.
Particles in suspension in a gravitational
field
Measure D and
we can find N.
31
Brownian motion
Einstein predicted thermal motions of
tiny particles visible under the
microscope and suspected that this
explained Brown’s observations of the
motion of pollen grains.
For particles of size 0.001mm, Einstein
predicted a displacement of approximately
6 microns in one minute.
“If it is really possible to observe the motion discussed here … then classical
thermodynamics can no longer be viewed as strictly valid even for
microscopically distinguishable spaces, and an exact determination of the
real size of atoms becomes possible.”
32
Estimating the coefficient of diffusion for suspended particles
…and hence determine N.
To describe the thermal motions of small particles, Einstein laid the foundations of the
modern theory of stochastic processes and solved the “random walk problem.”
Particles spread over time t,
distributed on a bell curve.
Their mean square
displacement is 2Dt.
Hence we can read D from
the observed displacement
of particles over time.
Then find N using
N = (RT/6pmD) x 1/P
33
Unexpected properties of the motion
Displacement is proportional to square
root of time. So an average velocity
cannot be usefully defined.
displacement  0 as time gets large.
time
The “jiggles” are not the visible result
of single collisions with water
molecules, but each jiggle is the
accumulated effect of many collision.
34
Thermodynamics of
Fluctuating Systems
of Independent
Components
35
Canonical Formulae
From Einstein’s
papers of 1902-1904
For a system of n spatially localized,
spatially independent components:
Probability
density over
states
Canonical
entropy

 E(p 1 ,..., p n ) 
p(x1 ,..., xn , p 1 ,..., p n )  exp 



kT
E
S   k ln
T



 E(p i ) 
 E(p i ) 
E
 exp  kT dp idxi  T  k ln  exp  kT dp i  dxi 
E
E
 k ln( JV n )   k ln J  nk ln V
T
T
!!!!!!!!!
Free energy

Pressure exerted
by components

F  kT ln
Energy E depends only
canonical momenta pi and
not on canonical positions xi.
terms dependent only
on momentum
degrees of freedom
Vn
 exp(E / kT )dpdx  kT ln J  nkT ln V
F 
d
nkT
P    
(nkT ln V ) 
V T dV
V
Ideal gas law
36
Macroscopic Signature of…
…a system of n spatially localized,
spatially independent components.
Sugar molecules in solution.
Microscopically visible corpuscles.
What else?
Entropy
varies logarithmically
with volume V
Pressure
obeys ideal gas law.
S=
terms in energy
and momentum
degrees of freedom
+ nk ln V
nkT
P
V
37
Einstein’s 1905 derivation of the ideal gas law
from the assumption of spatially independent, localized components
Brownian
motion
paper, §2
Osmotic
pressure
from the
viewpoint of
the
molecular
kinetic
theory of
heat.
38
The Miraculous
Argument
39
The Light Quantum Paper
40
The Light Quantum Paper
§1 On a difficulty encountered in the theory of “black-body
radiation”
§2 On Planck’s determination of the elementary quanta
§3 On the entropy of radiation
Development of
the “miraculous
argument”
§4 Limiting law for the entropy of monochromatic radiation at
low radiation density
§5 Molecular-theoretical investigation of the dependence of the
entropy of gases and dilute solutions on the volume
§6 Interpretation of the expression for the dependence of the
entropy of monochromatic radiation on volume according to
Boltzmann’s Principle
§7 On Stokes’ rule
Photoelectric
effect
§8 On the generation of cathode rays by illumination of solid
bodies
§9 On the ionization of gases by ultraviolet light
41
The Miraculous Argument. Step 1.
42
The Miraculous Argument. Step 1.
Probability that n
independently moving points
all fluctuate into a
subvolume v of volume v0
W = (v/v0
)n
e.g molecules in a
kinetic gas, solute
molecules in dilute
solution
Boltzmann’s Principle
S = k log W
Entropy change for the
fluctuation process
S - S0= kn log v/v0
Standard
thermodynamic
relations
Ideal gas law for kinetic
gases and osmotic pressure
of dilute solutions
Pv = nkT
43
The Miraculous Argument. Step 2.
44
The Miraculous Argument. Step 2.
Observationally derived
entropies of high frequency 
radiation of energy E and
volume v and v0
S - S0= k (E/h) log V/V0
Boltzmann’s Principle
S = k log W
Probability of constant
energy fluctuation in volume
from v to v0
W = (V/V0)E/h
Restate
in words
"Monochromatic radiation of low density behaves-as long as Wien's radiation formula is valid --in a
thermodynamic sense, as if it consisted of mutually
independent energy quanta of magnitude [h]."
45
A Familiar Project
46
The Light Quantum Paper
From
macroscopic
thermodynamic
properties of heat
radiation
infer
microscopic
constitution of
radiation.
47
Einstein’s Doctoral Dissertation
From
macroscopic
thermodynamics of
dilute sugar solutions
(viscosity, diffusion)
infer
microscopic
constitution
(size of sugar
molecules)
48
The “Brownian Motion” Paper
From
microscopically
visible motions of
small particles
infer
sub-microscopic
thermal motions of
water molecules and
vindicate the
molecular-kinetic
account.
49
The macroscopic signature of the
microscopic constitution of the light quantum paper
Find this dependence
macroscopically
Entropy change = k n log (volume ratio)
Infer the system consists
microscopically of n,
independent, spatially
localized points.
50
Complications
51
Einstein makes it
look too easy.
Just where is the signature?
entropy
density
Entropy of
volume V of heat
radiation at
frequency .
S() = s().V
Entropy is linear in V.
Pressure
energy
density
P = u/3
exerted by
radiation
Pressure is independent of V.
Ideal gas expanding isothermally
Heat radiation expanding isothermally
P  1/V
P is constant
Disanalogy:
expanding heat
radiation creates
new components.
n is constant.
P  n/V
Energy is constant.
n increases with V.
P  n/V
but n/v is constant.
Energy increases with n.
52
Canonical entropy
E
S   k ln J  nk ln V
T
Change in mean energy E
obscures ln V dependency.
53
Find a rare process of constant energy,
no new quanta created.
Radiation at
equilibrium state
occupies volume V0 .
fluctuates to
Momentary,
improbable compressed
state of volume V.
Constant energy.
Constant n.
DS = k ln (V/V0)
P  n/V  1/V
Logarithmic
dependency appears.
Ideal gas law appears
54
More
Complications
55
Canonical Entropy Formula of 1903…
A Theory of the
Foundations of
Thermodynamics,”
Annalen der Physik, 11
(1903), pp. 170-87.
…is briefly recapitulated in the
Brownian motion paper §2.
§6 On the Concept of Entropy
§7 On the Probability of
Distributions of State
§8 application of the Results to a
Particular Case
§9 Derivation of the Second Law
Canonical entropy for equilibrium
systems deduced from Clausius’ S  S0 

dqrev
T
56
…is inapplicable to the quanta of heat radiation
Phase space of fixed
(finite) dimensions
Fixed number of
components
Number of quanta is
variable
Definite equations of
motion in phase space
Equations of motion for
light quanta unknown
Miraculous argument assigns assigns entropy to
momentary, fluctuation states, far from equilibrium.
57
Einstein’s Demonstration
of Boltzmann’s Principle
S= k log W
58
The Demonstration
Probability W of two independent
states with probabilities W1 and W2
W = W1 x W2
Entropy S is function of W only
S = (W)
Entropies of independent
systems add
S = S1 + S2
S = const. log W
§5 light quantum paper.
Apparently avoids all the problems of
the canonical entropy formula.
59
Brilliant, but maddening!
Probability, in what probability space?
Probability W of two
independent states with
probabilities W1 and W2
W = W1 x W2
Entropy S is
function of W only
S = (W)
Entropies of
independent
systems add
S = S1 + S 2
S = const. log W
Is entropy a function of probability only?
Entropy S is defined so far for equilibrium states.
Is this a definition of the entropy
of non-equilibrium states?
…Boltzmann?
Connection to thermodynamic entropy?
dqrev
S

S

Clausius
0 
T
Entropy assigned is the entropy the
state would have if it were an
equilibrium state.

60
Conclusion
61
Goals…
The content of Einstein’s
discovery was quite unanticipated:
High frequency light energy exists in
• many,
• independent,
• spatially localized
points.
The method of Einstein’s discovery
was familiar and secure.
Einstein’s research program in statistical
physics from first publication of 1901:
How can we infer the microscopic
properties of matter from its macroscopic
properties?
Einstein’s visceral
mastery of fluctuations.
62
Finis
63
Appendices
64
Idea Gas Law
65
A much simpler derivation
Very many,
independent, small
particles at
equilibrium in a
gravitational field.
Pull of gravity
equilibrated by
pressure P.
Independence expressed:
energy E(h) of each particle
is a function of height h only.
Equilibration of pressure by a field instead of a semi-permeable membrane
was a device Einstein used repeatedly but casually in 1905, but had been
introduced with great caution and ceremony in his 1902 “Potentials” paper.
66
A much simpler derivation
Boltzmann
distribution of
energies
Probability of one molecule at height h
P(h) = const. exp(-E(h)/kT)
Density of gas at height h
r = r0 exp(-E(h)/kT)
Density gradient due to gravitational field
dr/dh = -1/kT (dE/dh) r = 1/kT f = 1/kT dP/dh
where f = - (dE/dh) r is the gravitational force
density, which is balanced by a pressure gradient P
for which f = dP/dh.
Rearrange
Ideal gas
law
So that
Reverse inference
possible, but
messy. Easier with
Einstein’s 1905
derivation.
d/dh(P - rkT) = 0
P = rkT
PV = nkT
since r= n/V
67
Ideal Gas Law
Microscopically…
many, independent,
spatially localized
points scatter due to
thermal motions
Pv = nkT
Macroscopically…
the spreading is
driven by a pressure
P =nkT/V
The equivialence was
standard. Arrhenius
(1887) used it as a
standard technique to
discern the degree of
dissociation of solutes
from their osmotic
pressure.
This equivalence was an essential component of Einstein’s
analysis of the diffusion of sugar in his dissertation and of the
scattering of small particles in the Brownian motion paper.
pressure driven
scattering
is balanced by
Stokes’ law
viscous forces
Relation between macroscopic diffusion
coefficient D and microscopic Avogadro’s
number N
D= (RT/6p viscosity) (1/N radius particle)
68
Ideal Gas Law Does Hold for Wien Regime Heat Radiation…
h 
8 p h 3
Wien
u( ,T ) 
exp


distribution
 kT 
c3
Full spectrum radiation
Radiation
pressure
Einstein, light
quantum paper, §6.
P
energy
=
density
u /3
mean energy
= 3kT
per quantum
=
nkT/V

Same result for single
frequency cut, but much
longer derivation!
energy density
= 3nkT/V
for n quanta
…but it is an unconvincing signature of discreteness
P = u/3 = T4/3 = (VT3/3k) k T/V = n kT/V
Heat radiation consists of n = (VT3/3k) localized
components, where n will vary with changes in
volume V and temperature T?
69