#### Einstein’s Miraculous Argument of 1905 John D. Norton Department of History and Philosophy of Science Center for Philosophy of Science University of Pittsburgh.

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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 - cc(r) for constants cand 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 4VT 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/6pmD) 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/6pmD) 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/6pmD) 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/6pmD) 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(p i ) E(p i ) E E S k ln exp dp dx k ln exp dp i dxi kT i i T T kT E E k ln(JV n ) k ln J nklnV T T !!!!!!!!! Free energy Pressure exerted by components Energy E depends only canonical momenta pi and not on canonical positions xi. terms dependent only on momentum degrees of freedom Vn F kT ln exp(E / kT)dpdx kT ln J nkT lnV F d nkT P (nkT lnV ) 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 lnV 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 8ph 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