1. Atoms Entropy Quanta John D. Norton
Department of History and Philosophy of Science
University of Pittsburgh

2. The Papers of Einstein’s Year of Miracles, 1905

3. The Papers of 1905 1
"Light quantum/photoelectric effect paper"
"On a heuristic viewpoint concerning the production and transformation of light."
Annalen der Physik, 17(1905), pp (17 March 1905)
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. 2
Einstein's doctoral dissertation
"A New Determination of Molecular Dimensions"
Buchdruckerei K. J. Wyss, Bern, (30 April 1905)
Also: Annalen der Physik, 19(1906), pp
Einstein used known physical properties of sugar solution (viscosity, diffusion) to determine the size of sugar molecules.
"Brownian motion paper."
"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 (May 1905; received 11 May 1905)
Einstein predicted that the thermal energy of small particles would manifest as a jiggling motion, visible under the microscope. 3 4
Special relativity
“On the Electrodynamics of Moving Bodies,”
Annalen der Physik, 17 (1905), pp (June 1905; received 30 June, 1905)
Maintaining the principle of relativity in electrodynamics requires a new theory of space and time.
E=mc2
“Does the Inertia of a Body Depend upon its Energy Content?”
Annalen der Physik, 18(1905), pp (September 1905; received 27 September, 1905)
Changing the energy of a body changes its inertia in accord with E=mc2. 5

4. Einstein to Conrad Habicht
18th or 25th May 1905

5. “…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.

6. Why is only the light quantum “very revolutionary”?
All the rest develop or complete 19th century physics. 2
Einstein's doctoral dissertation
"A New Determination of Molecular Dimensions"
Buchdruckerei K. J. Wyss, Bern, (30 April 1905)
Also: Annalen der Physik, 19(1906), pp
Advances the molecular kinetic program of Maxwell and Boltzmann.
"Brownian motion paper."
"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 (May 1905; received 11 May 1905) 3 4
Special relativity
“On the Electrodynamics of Moving Bodies,”
Annalen der Physik, 17 (1905), pp (June 1905; received 30 June, 1905)
Establishes the real significance of the Lorentz covariance of Maxwell’s electrodynamics.
E=mc2
“Does the Inertia of a Body Depend upon its Energy Content?”
Annalen der Physik, 18(1905), pp (September 1905; received 27 September, 1905) 5
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.
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.
"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]."
Einstein’s light quantum paper initiated a reappraisal of the physical constitution of light that is not resolved over 100 years later.

8. How did Einstein find the quantum?
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)

9. 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.

10. Einstein’s Early Program in Statistical Physics

11. 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
“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

12. Einstein’s first two “worthless” papers
Einstein’s hypothesis:
Forces between molecules at distance r apart are governed by a potential P satisfying
P = P - cc(r)
for constants cand c characteristic of the two molecules and universal function (r).
macroscopic properties of capillarity and electrochemical potentials
infer
coefficients in the microscopic force law.
From

13. Independent Discovery of the Gibbs Framework
3 papers
Einstein, Albert.
'Kinetische Theorie des Waermegleichgewichtes und des zweiten Hauptsatzes der Thermodynamik'. Annalen der Physik, 9 (1902)

14. Independent Discovery of the Gibbs Framework
3 papers
Einstein, Albert. 'Eine Theorie der Grundlagen der Thermodynamik'. Annalen der Physik, 9 (1903)

15. Independent Discovery of the Gibbs Framework
3 papers
Einstein, Albert. 'Zur allgemeinen molekularen Theorie der Waerme'. Annalen der Physik, 14 (1904)

16. The Hidden Gem

17. Einstein’s Fluctuation Formula
Any canonically distributed system
Variance of energy from mean
Heat capacity is
macroscopically measureable.
(1904)

18. Applied to an Ideal Gas Ideal monatomic gas, n molecules
rms deviation of energy from mean
n=1024…. negligible
n=1

19. Applied to heat radiation
In 1904, no one had any solid idea of the constitution of heat radiation!
Volume V of heat radiation (Stefan-Boltzmann law)
rms deviation of energy from mean
Hence estimate volume V in which fluctuations are of the size of the mean energy.
(1904)

20. Einstein’s Doctoral Dissertation "A New Determination of Molecular Dimensions”

22. How big are molecules. = How many fit into a gram mole
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 j)
= (r/m) N (4p/3) P 3
= sugar density in the solution
m = molecular weight of sugar
Well, not quite. Einstein made a mistake in the calculation. The correct result is
m* = m (1 + 5/2 j)
The examiners did not notice. Einstein passed and was awarded the PhD. He later corrected the mistake.
P = radius of sugar molecule, idealized as a sphere

23. Recovering N N = (3m/ 4pr) x (m*/m - 1) x 1/P 3 N = (RT/6pm D) x 1/P
Turning the expression for apparent viscosity inside out:
P = radius of molecule. Unknown!
N = (3m/ 4pr) x (m*/m - 1) x 1/P 3
All measurable quantities
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

24. 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.

25. 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
The condition for perfect balance is
N = (RT/6pm D) x 1/P
And
Sugar molecules diffusing
upwards because of the concentration gradient.
density gradient
pressure gradient (ideal gas law)
upward force

26. 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.

27. “Brownian motion paper
“Brownian motion paper.” “On the motion of small particles suspended in liquids at rest required by the molecular-kinetic theory of heat.”

29. 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!
…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
The particles exert a pressure due to their thermal motions.
PV=nkT
Measure D and we can find N.

30. 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.”

31. 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

32. Unexpected properties of the motion
The “jiggles” are not the visible result of single collisions with water molecules, but each jiggle is the accumulated effect of many collision.
Displacement is proportional to square root of time. So an average velocity cannot be usefully defined.
displacement
time
 0 as time gets large.

33. Thermodynamics of Systems of Independent Components (ideal gas law)

34. The simplest case: an ideal gas
(e.g. most ordinary gases at ordinary temperatures and low pressures)
Pressure P x volume V =
number of molecules n
x Boltzmann’s constant k
x temperature T
PV = nkT
The remarkable fact: Very few micro-properties of the gas enter into the law. Only n and k.
What we shall see:
The law depends only the gases having a very few simple properties: they consist of very many spatially localized, independent components, fixed in number.
Boltzmann’s constant k
= ideal gas constant R
/Loschmidt’s number N
k = R/N

35. From micro to macro… Micro-structure
A dilute gas is a fixed, large number of component molecules that do not interact with each other.
An ideal gas in a gravitational field.
Because of their thermal energy at temperature T,
the molecules spread to fill the volume V,
exerting a pressure P on the vessel walls. h
Observed property
Ideal gas law PV = nkT
We can infer the ideal gas law from the micro-structure.

36. …and back An ideal gas in a gravitational field. Micro-structure
A dilute gas is a fixed, large number of component molecules that do not interact with each other.
An ideal gas in a gravitational field.
The ideal gas law
is the macroscopic
signature of… h
Observed property
Ideal gas law PV = nkT
…And we can invert the inference.

37. 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.

38. 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
 = 0 exp(-E(h)/kT)
Reverse inference possible, but messy. Easier with Einstein’s 1905 derivation.
Density gradient due to gravitational field
d/dh = -1/kT (dE/dh)  = 1/kT f = 1/kT dP/dh
where f = - (dE/dh)  is the gravitational force density, which is balanced by a pressure gradient P for which f = dP/dh.
Rearrange d/dh(P - kT) = 0
So that P = kT PV = nkT
since = n/V
Ideal gas law

39. “Light quantum/photoelectric effect paper” "On a heuristic viewpoint concerning the production and transformation of light."

40. Einstein’s astonishing idea
The light quantum hypothesis
“Monochromatic radiation of low density (within the range of validity of Wien’s radiation formula) behaves thermodynamically as if it consisted of mutually independent energy quanta of magnitude [hn].”
i.e. Electromagnetic radiation does not always behave like waves, contrary to the most successful science of the nineteenth century. Sometimes it behaves like little lumps of energy of size hn.
Einstein’s reasons:
Certain experimental effects are only plausibly explained by light consisting of quanta: especially the photoelectric effect.
There is an atomistic signature in the thermodynamics of high frequency radiation.

41. Why the atomistic signature of radiation is hard to see
High frequency radiation is a system with
• variable, large number
• spatially localized
• independent
components, so the ideal gas law is hard to see.
Constant temperature expansion of high frequency radiation. The number of quanta increases and the pressure stays constant.
Atomistic signature of systems with
• fixed, large number
• spatially localized
• independent
components is the ideal gas law.
Constant temperature expansion of an ideal gas. The number of molecules is constant and the pressure drops, as the ideal gas law demands.

42. Einstein noticed a new signature of independent atoms…
An ideal gas can spontaneously compress to half its volume (with very small probability).
The signature of independent atoms: entropy depends on the logarithm of volume ratios.
Entropy difference
DS = - kn log 2
Probability one molecule is on the left is (1/2).
Probability all n molecules are on the left is (1/2)n.
Entropy S =
- k log (probability)
Entropy difference
DS = - kn log 2

43. …which he found in high frequency radiation
Spontaneous volume fluctuations in high frequency radiation are also possible.
Radiation of energy E may spontaneously halve in volume with very small probability.
log (probability)
= - Entropy s/k
Probability of spontaneous compression is (1/2)E/hn
Determined from measured properties of radiation.
Entropy difference
DS = - k (E/hn) log 2
Just as is the radiation consisted of n = E/hn spatially localized, independent quanta of energy.

44. Conclusion

45. Unity in Einstein’s statistical physics of 1905
Einstein investigated many different thermal systems in 1905.
What made these investigations tractable was a common feature:
Each consisted of a large number of spatially localized component that do not interact with each other. And so:
Dilute
sugar solution
Particles
in suspension
High
frequency
radiation
Ideal gas
is just like
is just like
is just like