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Einstein’s Miraculous

Argument of 1905John D. Norton

Department of History and Philosophy of ScienceCenter for Philosophy of Science

University of Pittsburgh

Win $$ from your friends…

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Q.For which work, published in 1905, was Albert Einstein awarded his doctoral degree?

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Warm UpThe Papers of Einstein’s

Year of Miracles, 1905

The Papers of 1905

Einstein's doctoral dissertation"A New Determination of Molecular Dimensions"Buchdruckerei K. J. Wyss, Bern, 1905. (30 April 1905)Also: Annalen der Physik, 19(1906), pp. 289-305.

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. 549-560.(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.

"Light quantum/photoelectric effect paper""On a heuristic viewpoint concerning the production and transformation of light." Annalen der Physik, 17(1905), pp. 132-148.(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.

Special relativity“On the Electrodynamics of Moving Bodies,”Annalen der Physik, 17 (1905), pp. 891-921. (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. 639-641. (September 1905; received 27 September, 1905)

Changing the energy of a body changes its inertia in accord with E=mc2.

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Einstein to Conrad Habicht18th or 25th May 1905

“…and is very revolutionary”

…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.

Einstein’s assessment of his light quantum paper.

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Advances the molecular kinetic program of Maxwell and Boltzmann.

Establishes the real significance of the Lorentz covariance of Maxwell’s electrodynamics.

Light energy has momentum; extend to all forms of energy.

Why is only the light quantum“very revolutionary”?

All the rest develop or complete 19th century physics.

Einstein's doctoral dissertation"A New Determination of Molecular Dimensions"Buchdruckerei K. J. Wyss, Bern, 1905. (30 April 1905)Also: Annalen der Physik, 19(1906), pp. 289-305.

"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. 549-560.(May 1905; received 11 May 1905)

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3Special relativity“On the Electrodynamics of Moving Bodies,”Annalen der Physik, 17 (1905), pp. 891-921. (June 1905; received 30 June, 1905)

E=mc2

“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)

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Why is only the light quantum“very revolutionary”?

All the rest develop or complete 19th century physics.

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.

Well, not always.

Einstein’s light quantum paper initiated a reappraisal of the physical constitution of light that is not resolved over 100 years later.

"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]."

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Goalsof this presentation

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Historical-methodological…The content of Einstein’s discovery was quite unanticipated:

High frequency light energy exists in• spatially independent,• spatially localizedpoints.

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)

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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.

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….

Foundational…

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Einstein’s visceral mastery of

fluctuations.

Why we need atoms and not just ordinary thermodynamics.

How to compute fluctuations.“Boltzmann’s Principle.”The thermodynamics

of fluctuating, few component systems.

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Einstein’s Early Program in Statistical Physics

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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. 513-523.

“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.

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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 - cc(r)

for constants cand 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

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.

Independent Discovery of the Gibbs Framework

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Einstein, Albert.'Kinetische Theorie des Waermegleichgewichtes und des zweiten Hauptsatzes der Thermodynamik'. Annalen der Physik, 9 (1902)

3 papers 1902-1904

Independent Discovery of the Gibbs Framework

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Einstein, Albert. 'Eine Theorie der Grundlagen der Thermodynamik'. Annalen der Physik, 9 (1903)

3 papers 1902-1904

Independent Discovery of the Gibbs Framework

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Einstein, Albert. 'Zur allgemeinen molekularen Theorie der Waerme'. Annalen der Physik, 14 (1904)

3 papers 1902-1904

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The Hidden Gem

€

ε 2 = (E − E)2 = kT 2 dEdT

= kT 2C

Variance of energy from mean

Einstein’s Fluctuation Formula

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€

p(E)∝ exp −EkT

⎛ ⎝ ⎜

⎞ ⎠ ⎟

Any canonically distributed system

(1904)

Heat capacity ismacroscopically

measureable.

Applied to an Ideal Gas

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Ideal monatomic gas, n molecules

€

E =3nkT

2

€

C =3nk2

€

ε2 = (E − E)2 = kT 2 dEdT

= kT 2C

rms deviation of energy from mean

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εE

=3n / 2kT

(3n / 2)kT=

13n / 2

n=1024…. negligible

n=1

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13/2

= 0.816

Applied to heat radiation

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Volume V of heat radiation (Stefan-Boltzmann law)

€

E =σVT 4

€

C = 4σVT 3

€

ε2 = (E − E)2 = kT 2 dEdT

= kT 2C

rms deviation of energy from mean

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εE

=2 σkVT 5/2

σVT 4 = 2k

σV1

T 3/2

In 1904, no one had any solid idea of the constitution of heat radiation!

Hence estimate volume V in which fluctuations are of the size of the mean energy.

€

ε2 = E2

(1904)

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Einstein’s Doctoral Dissertation

"A New Determination of Molecular Dimensions”

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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…

P = radius of sugar molecule, idealized as a sphere

Well, not quite. Einstein made a mistake in the calculation. The correct result is* = (1 + 5/2 )The examiners did not notice. Einstein passed and was awarded the PhD. He later corrected the mistake.

Apparent viscosity *

=viscosity of pure water x (1 + fraction of volume taken by sugar )

= (/m) N (4/3) P 3= sugar density in the solutionm = molecular weight of sugar

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All measurable quantities

TWO equations in TWO unknowns.

P = radius of molecule. Unknown!

All measurable quantities

Recovering N

Turning the expression for apparent viscosity inside out:

N = (3m/ 4) x (*/ - 1) x 1/P 3

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/6D) x 1/P

Einstein determinedN = 2.1 x 10 23

After later correction for his calculation error

N = 6.6 x 10 23

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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 = nkTDilute sugar solution in a gravitational field.

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Recovering the equation for diffusionThe equilibrium sugar concentration gradient arises from a balance of:

Sugar molecules fallingunder the effect of gravity.Stokes’ law F = 6Pv, F = gravitational force

The condition for perfect balance is

N = (RT/6D) x 1/P

And

Sugar molecules diffusingupwards because of the concentration gradient.

density gradient

pressure gradient

(ideal gas law)

upward force

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

kinetic theory of heat.”

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An easier way to estimate N?

Doctoral dissertation: Einstein determined N from TWO equations in TWO unknowns N, P.

N = (3m/ 4) x (*/ - 1) x 1/P 3 N = (RT/6D) x 1/PWhat 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/6D) x 1/P.

Particles in suspension in a gravitational field

The particles exert a pressure due to their thermal motions.

PV=nkT

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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.”

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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/6D) x 1/P

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Unexpected properties of the motion

Displacement is proportional to square root of time. So an average velocity cannot be usefully defined.

0 as time gets large.

The “jiggles” are not the visible result of single collisions with water molecules, but each jiggle is the accumulated effect of many collision.

displacementtime

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Thermodynamics of Fluctuating Systems

of Independent Components

terms dependent only on momentum

degrees of freedomVn

!!!!!!!!!

Canonical Formulae

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From Einstein’s papers of 1902-1904

For a system of n spatially localized, spatially independent components:

Probability density over

states

€

p(x1,..., xn,π1,...,π n )∝ exp −E(π1,...,π n )

kT ⎛ ⎝ ⎜

⎞ ⎠ ⎟

Energy E depends only canonical momenta i and not on canonical positions xi.

Canonical entropy

€

S =ET

+ k ln exp −E(π i )

kT ⎛ ⎝ ⎜

⎞ ⎠ ⎟∫ dπ idxi =

ET

+ k ln exp −E(π i )

kT ⎛ ⎝ ⎜

⎞ ⎠ ⎟∫ dπ i dxi∫

⎛ ⎝ ⎜

⎞ ⎠ ⎟

=ET

+ k ln(JV n ) =ET

+ k ln J + nk lnV

Free energy

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F = −kT ln exp(−E / kT )∫ dπdx = −kT ln J − nkT lnV

Pressure exerted by components

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P = −∂F∂V ⎛ ⎝ ⎜

⎞ ⎠ ⎟

T

=d

dV(nkT lnV ) =

nkTV

Ideal gas law

Macroscopic Signature of…

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…a system of n spatially localized, spatially independent components.

Pressureobeys ideal gas law.

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P =nkTV

Sugar molecules in solution.Microscopically visible corpuscles.What else?

Entropy varies logarithmically

with volume VS = terms in energy

and momentum degrees of freedom

+ nk ln V

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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.

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The Miraculous Argument

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The Light Quantum Paper

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Development of the “miraculous

argument”

Photoelectric effect

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

§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

§8 On the generation of cathode rays by illumination of solid bodies

§9 On the ionization of gases by ultraviolet light

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The Miraculous Argument. Step 1.

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The Miraculous Argument. Step 1.

Probability that n independently moving points

all fluctuate into a subvolume v of volume v0

W = (v/v0)ne.g molecules in a kinetic gas, solute molecules in dilute solution

Boltzmann’s PrincipleS = 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 solutionsPv = nkT

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The Miraculous Argument. Step 2.

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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

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]."

W = (V/V0)E/h

Boltzmann’s PrincipleS = k log W

Probability of constant energy fluctuation in volume from v

to v0

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A Familiar Project

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The Light Quantum Paper

macroscopic thermodynamic properties of heat radiation

infer

microscopic constitution of radiation.

From

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Einstein’s Doctoral Dissertation

macroscopic thermodynamics of dilute sugar solutions (viscosity, diffusion)

infer

microscopic constitution(size of sugar molecules)

From

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The “Brownian Motion” Paper

microscopically visible motions of small particles

infer

sub-microscopic thermal motions of water molecules and vindicate the molecular-kinetic account.

From

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Infer the system consists microscopically of n, independent, spatially localized points.

The macroscopic signature of the microscopic constitution of the light quantum paper

Find this dependence macroscopically

Entropy change = k n log (volume ratio)

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Complications

Just where is the signature?

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Entropy of volume V of heat

radiation at frequency .

S() = s().VPressure exerted by

radiationP = u/3

Entropy is linear in V. Pressure is independent of V.

P n/V but n/v is constant.P n/VEnergy is constant. Energy increases with n.

Ideal gas expanding isothermally

P 1/V

n is constant.

Heat radiation expanding isothermallyP is constant

Disanalogy: expanding heat radiation creates new components.

n increases with V.

entropy density

energy density

Einstein makes it look too easy.

Change in mean energy E obscures ln V dependency.

Canonical entropy

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€

S =ET

+ k ln J + nk lnV

Find a rare process of constant energy,no new quanta created.

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Radiation at equilibrium state

occupies volume V0 .

fluctuates toMomentary, improbable compressed state of volume V.

Constant energy.Constant n.

S = k ln (V/V0)Logarithmic

dependency appears.

P n/V 1/VIdeal gas law appears

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MoreComplications

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Canonical Entropy Formula of 1903…

A Theory of the Foundations of Thermodynamics,” Annalen der Physik, 11 (1903), pp. 170-87.

§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

…is briefly recapitulated in the Brownian motion paper §2.

€

S − S0 = dqrevT

∫Canonical entropy for equilibrium systems deduced from Clausius’

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…is inapplicable to the quanta of heat radiation

Definite equations of motion in phase space

Equations of motion for light quanta unknown

Number of quanta is variable

Phase space of fixed (finite) dimensions

Fixed number of components

Miraculous argument assigns assigns entropy to momentary, fluctuation states, far from equilibrium.

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Einstein’s Demonstration of Boltzmann’s Principle

S= k log W

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The Demonstration Probability W of two independent

states with probabilities W1 and W2

W = W1 x W2

S = const. log W

Entropy S is function of W onlyS = (W)

Entropies of independent systems add

S = S1 + S2

§5 light quantum paper.Apparently avoids all the problems of the canonical entropy formula.

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Brilliant, but maddening!

Probability W of two independent states with probabilities W1 and W2

W = W1 x W2

S = const. log W

Entropy S is function of W onlyS = (W)

Entropies of independent systems add

S = S1 + S2

Probability, in what probability space?

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?

Clausius

€

S − S0 = dqrevT∫

Connection to thermodynamic entropy?

Entropy assigned is the entropy the state would have if it were an equilibrium state.

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Conclusion

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Goals…The content of Einstein’s discovery was quite unanticipated:

High frequency light energy exists in• many,• independent,• spatially localizedpoints.

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.

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Finis

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Appendices

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Idea Gas Law

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A much simpler derivation

Very many, independent, small particles at equilibrium in a gravitational field.

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.

Independence expressed: energy E(h) of each particle is a function of height h only.

Pull of gravity equilibrated by pressure P.

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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)

Density gradient due to gravitational field d/dh = -1/kT (dE/dh) = 1/kT f = 1/kT dP/dhwhere 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 = nkTsince = n/V

A much simpler derivationBoltzmann distribution of energies

Ideal gas law

Reverse inference possible, but messy. Easier

with Einstein’s 1905 derivation.

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Ideal Gas Law Pv = nkT

Microscopically…many, independent, spatially localized points scatter due to thermal motions

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.

Stokes’ law viscous forces

pressure driven scattering

is balanced by

Relation between macroscopic diffusion coefficient D and microscopic Avogadro’s number ND= (RT/6 viscosity) (1/N radius particle)

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Heat radiation consists of n = (VT3/3k) localized components, where n will vary with changes in volume V and temperature T?

= n kT/V

Ideal Gas Law Does Hold for Wien Regime Heat Radiation…

Wien distribution

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u(ν ,T ) = 8πhν 3

c3 exp −hνkT

⎛ ⎝ ⎜

⎞ ⎠ ⎟

Radiation pressure =

energy density u /3

mean energy per quantum

= 3kTenergy density

for n quanta = 3nkT/V

= nkT/V

Einstein, light quantum paper, §6.

PFull spectrum radiation

Same result for single frequency cut, but much longer derivation!

…but it is an unconvincing signature of discreteness

P = u/3 = T4/3 = (VT3/3k) k T/V