Einstein’s Miraculous Argument of 1905

1. Einstein’s Miraculous Argument of 1905 • John D. Norton • Department of History and Philosophy of Science • University of Pittsburgh

2. Thermodynamics of Fluctuating Systems of Independent Components

3. Einstein’s 1905 derivation of the ideal gas lawfrom the assumption of spatially independent, localized components • Brownian motion paper, §2 Osmotic pressure from the viewpoint of the molecular kinetic theory of heat.

4. Einstein’s 1905 derivation of the ideal gas lawfrom the assumption of spatially independent, localized components Canonical Formulae From Einstein’s papers of 1902-1904 For a system of n spatially localized, spatially independent components: Energy E depends only canonical momenta pi and not on canonical positions xi. Probability density over states Canonical entropy terms dependent only on momentum degrees of freedom Vn !!!!!!!!! Free energy Pressure exerted by components Ideal gas law

5. 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 S = terms in energy and momentum degrees of freedom + nk ln V Pressure obeys ideal gas law.

6. The Miraculous Argument

7. The Light Quantum Paper

8. 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

9. The Miraculous Argument. Step 1.

10. 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 The Miraculous Argument. Step 1. Probability that n independently moving points all fluctuate into a subvolume v of volume v0 e.g molecules in a kinetic gas, solute molecules in dilute solution W = (v/v0)n

11. The Miraculous Argument. Step 2.

12. 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]." The Miraculous Argument. Step 2.

13. A Familiar Project

14. The Light Quantum Paper From macroscopic thermodynamic properties of heat radiation infer microscopic constitution of radiation.

15. Einstein’s Doctoral Dissertation From macroscopic thermodynamics of dilute sugar solutions (viscosity, diffusion) infer microscopic constitution (size of sugar molecules)

16. 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.

17. 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)

18. Complications

19. Einstein makes it look too easy. Just where is the signature? entropy density Entropy of volume V of heat radiation at frequency n. energy density Pressure exerted by radiation S(n) = s(n).V P = u/3 Entropy is linear in V. Pressure is independent of V. Heat radiation expanding isothermally P is constant Ideal gas expanding isothermally P  1/V Disanalogy: expanding heat radiation creates new components. n increases with V. n is constant. P  n/V P  n/V but n/v is constant. Energy is constant. Energy increases with n.

20. Canonical entropy Change in mean energy E obscures ln V dependency.

21. Find a rare process of constant energy,no new quanta created. Momentary, improbable compressed state of volume V. Radiation at equilibrium state occupies volume V0 . fluctuates to Constant energy. Constant n. DS = k ln (V/V0) P  n/V  1/V Logarithmic dependency appears. Ideal gas law appears

22. MoreComplications

23. …is briefly recapitulated in the Brownian motion paper §2. Canonical entropy for equilibrium systems deduced from Clausius’ 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

24. Fixed number of components Number of quanta is variable Phase space of fixed (finite) dimensions Definite equations of motion in phase space Equations of motion for light quanta unknown …is inapplicable to the quanta of heat radiation Miraculous argument assigns assigns entropy to momentary, fluctuation states, far from equilibrium.

25. Einstein’s Demonstration of Boltzmann’s Principle S= k log W

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

27. Brilliant, but maddening! Probability, in what probability space? Probability W of two independent states with probabilities W1 and W2 Is entropy a function of probability only? W =W1 x W2 Entropy S is defined so far for equilibrium states. Entropy S is function of W only S = (W) Entropies of independent systems add S =S1 + S2 Is this a definition of the entropy of non-equilibrium states? …Boltzmann? Connection to thermodynamic entropy? Clausius S = const. log W Entropy assigned is the entropy the state would have if it were an equilibrium state.

28. Finis

29. Appendices

30. 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 Relation between macroscopic diffusion coefficient D and microscopic Avogadro’s number N D= (RT/6 viscosity) (1/N radius particle) is balanced by Stokes’ law viscous forces Ideal Gas Law Pv = nkT The equivialence was standard. Arrhenius (1887) used it as a standard technique to discern the degree of dissociation of solutes from their osmotic pressure. Microscopically… many, independent, spatially localized points scatter due to thermal motions Macroscopically… the spreading is driven by a pressure P =nkT/V

31. = nkT/V energy density for n quanta mean energy per quantum Einstein, light quantum paper, §6. = 3nkT/V = 3kT …but it is an unconvincing signature of discreteness = n kT/V P = u/3 = T4/3 = (VT3/3k) k T/V Heat radiation consists of n = (VT3/3k) localized components, where n will vary with changes in volume V and temperature T? Ideal Gas Law Does Hold for Wien Regime Heat Radiation… Wien distribution Full spectrum radiation Same result for single frequency cut, but much longer derivation! energy density Radiation pressure P u /3 =