Einstein’s approach to Statistical Mechanics A prelude to the Marvelous Year Luca Peliti In collaboration with Raúl Rechtman (UNAM, Temixco, México) May 30, 2018 Statistical Physics, SISSA and SMRI (Italy) 1
Outline Introduction The Papers Einstein vs. Gibbs Summary 2
Einstein before Einstein? • Einstein’s approach to Statistical Mechanics is independent and bolder than Gibbs’ • Einstein focuses on fluctuations as a tool for discovery, rather than a nuisance • The search for observable fluctuations leads him to focus on black-body radiation 3
The Papers I. Kinetic theory of thermal equilibrium and of the second law of thermodynamics II. A theory of the foundations of thermodynamics III. On the general molecular theory of heat 4
Some biographical facts • In 1902 Einstein had left the ETH having obtained a diploma in 1900, but not the doctorate • In spring 1902 his application for Technical Assistant, 3rd Class, to the Federal Office for Intellectual Property in Bern was accepted, and he started working there in June • He married Mileva Marić, whom she had met as a fellow student at ETH, in January 1903. Their first son was born in May 1904 • Before the three papers which interest us, he had published two papers in Annalen der Physik , which he much later judged “worthless” 5
Atomism in the XIX Century • Chemists: Dalton, Avogadro, Cannizzaro The atomic idea becomes a scientific tool • Early kinetic theory: Herapath, Waterston Forgotten for lack of observable consequences • Kinetic theory: Clausius, Maxwell, Boltzmann, Loschmidt Maxwell: gas viscosity does not depend on density Connections with thermodynamics, the problem of entropy • “Energetists” (e.g., Ostwald and Mach): Atoms are a concept and a calculating tool, not a reality 6
The man who trusted atoms • 1870: Ergodic hypothesis and physical interpretation of the temperature • 1872: Boltzmann’s equation and the H -theorem • 1877: S = k B log W and the Boltzmann distribution for “complex molecules” (in a footnote to Gastheorie he claims that it can be extended to arbitrary bodies) • 1884: Microcanonical and Canonical ensembles (respectively called monode and holode ) 7
Einstein’s motivations • Einstein aims to “derive the postulates of thermal equilibrium and the second principle using exclusively the mechanical equations and the probability calculus” • He mentions that he wishes to “fill in the gap” left by Maxwell and Boltzmann, “although [their] theories have come close to this goal” • He provides “a generalization of the second principle, which is useful for the application of thermodynamics” • He also gives the “mathematical expression of entropy from a mechanical point of view” • The 1902–03 papers have similar structure: I’ll deal with them in one go 8
Mechanical description • General description of a mechanical system: dp i dt = ϕ i ( p 1 , . . . , p n ) • Energy is the unique integral of motion: E ( p 1 , . . . , p n ) = const . • (Liouville’s theorem is assumed in 1902, only implicitly in 1903): ∂ϕ i ∑ = 0 ∂p i i 9
Probabilistic description • Observable quantities are given by temporal averages of functions of state variables: ∫ T 1 A = lim dt A ( p 1 ( t ) , . . . , p n ( t )) T T →∞ 0 • For a given value of E , all observable quantities take on a constant value at equilibrium • Ergodic hypothesis : for any region Γ of state space, let τ be the time spent in Γ during time T . Then ∫ τ lim T = const . = ϵ ( p 1 , . . . , p n ) dp 1 · · · dp n T →∞ Γ 10
Probabilistic description • Ensemble : Given N systems of the same type, the number dN of systems in the small region g at any given time is ∫ dN = ϵ ( p 1 , . . . , p n ) dp 1 · · · dp n g • From stationarity (and Liouville’s theorem) one obtains ϵ ( p 1 , . . . , p n ) = const . • Einstein has thus derived the microcanonical ensemble 10
Canonical ensemble • Consider a small system σ in contact with a much larger one Σ with total energy E ∗ ≤ E ≤ E ∗ + δE ∗ E = η + H, • Consider g : π i ≤ π i ≤ π i + δπ i ( = 1 , . . . , ℓ ) and G : Π i ≤ Π i ≤ Π i + δ Π i ( i = 1 , . . . λ ): • dN 1 : number of systems that are found in g × G : dN 1 = C · dπ 1 · · · dπ ℓ d Π 1 · · · d Π λ = const . e − 2 h ( H + η ) dπ 1 · · · dπ ℓ d Π 1 · · · d Π λ • Number of systems for which the state of σ lies in g : ∫ dN ∝ e − 2 hη dπ 1 · · · dπ ℓ e − 2 hH d Π 1 · · · d Π λ E ∗ − η ≤ H ≤ E ∗ + δE ∗ − η 11
Canonical ensemble ∫ E ≤ H ≤ E + δE e − 2 hH d Π 1 · · · d Π λ ≃ e − 2 hE ω ( E ) • χ ( E ) = • Choosing h such that 2 h = ω ′ ( E ) ω ( E ) χ is independent of the state of σ and we have dN = const . e − 2 hη dπ 1 · · · dπ ℓ • The system σ acts like a thermometer, and if σ 1 and σ 2 are each in equilibrium with Σ , they are in equilibrium with each other (“0-th principle”) • Choosing σ as a single molecule, its average energy is equal to 3 / 4 h and thus 2 h = 1 /k B T (in modern notation) 11
The Entropy Infinitely slow processes • Einstein considers two kinds of transformations: Adiabatic transformations: the evolution equations hold at every time, but the ϕ i ’s can vary by external action via parameters λ “Isopycnic” (=equal-density) transformations: correspond to the thermal contact with a body at a different temperature: the evolution equations do not hold during the transformation, but before and after • Any infinitely slow process can be approximated by a succession of adiabatic and isopycnic transformations 12
The Entropy • During an infinitely slow process one has ∑ ∂E ∂E ∑ dE = ∂λ dλ + dp ν ∂p ν ν � �� � dQ • The canonical distribution holds before and after an infinitesimal transformation, thus from dW = e c − 2 hE dp 1 · · · dp n one obtains from the normalization of W ∫ e c + dc − 2( h + dh )( E + ∑ ∂ λ E dλ ) dp 1 · · · dp n = 0 (neglecting fluctuations in E ) leading to 2 h dQ = d (2 hE − c ) 12
The Entropy Thus, since 1 / 4 h = κT (kinetic energy of a degree of freedom) ( E ) dS = dQ = d T − 2 κc T leading to ∫ S = E e − 2 hE dp 1 · · · dp n T + 2 κ log (As far as I know Boltzmann did not have such a general expression) But what about ∆ S ≥ 0 ? 12
On the growth of entropy Even Einstein has some difficulties with entropy growth… • Consider an ensemble of N systems of energy between E and E + δE , and divide the available phase space into regions g k of equal volume • Define a “state” by assigning the number N k of systems which lie in g k • Define the “probability” W of a state as the number of ways of distributing the systems compatible with the state. One has ∫ N ! N 1 ! · · · N k ! · · · ≃ const. − log W = log ρ t log ρ t dp 1 · · · dp n • Then W is maximal when the distribution is uniform 13
On the growth of entropy • “We have to assume” that W never decreases: thus ∫ ∫ for t ′ ≥ t ρ t ′ log ρ t ′ dp 1 · · · dp n ≥ − ρ t log ρ t dp 1 · · · dp n − • From this Einstein deduces (!) that − log ρ t ′ ≥ − log ρ t (again neglecting fluctuations…) • Consider a collection of systems σ 1 , σ 2 , . . . initially isolated and let them exchange heat among themselves, then get isolated again and reach equilibrium ν c ν − 2 h ν E ν ∏ dp evolves • The initial state dw = dw 1 · dw 2 · · · = e ∑ ν ∏ dp into the final state dw ′ = dw ′ ∑ ν c ′ ν − 2 h ′ ν E ′ 1 · dw ′ 2 · · · = e • Thus from ρ t ′ ≤ ρ t one obtains ∑ ν ≤ ∑ ν c ′ ν − 2 h ′ ν E ′ ν c ν − 2 h ν E ν , i.e., ∑ ν ≥ ∑ ν S ′ ν S ν 13
The “Gap” What was the “gap” Einstein wished to fill? There is some debate • Boltzmann had introduced his ensembles ( holode and ergode ) as mechanical examples of systems satisfying statistical mechanics • Einstein considers them as description of actual physical systems (and it is not known how much he knew about them) • Einstein probably felt that his derivation of equipartition was more general than Boltzmann’s • Renn has argued from hints in a letter by Marić that equipartition was at the center of Einstein thoughts in 1901–02 14
The 1904 Paper • New expression from the entropy: given ∫ E<E ( p ) <E + δE dp one has ω ( E ) δE = ∫ dE ∫ ω ′ ( E ) S = = 2 κ ω ( E ) dE = 2 κ log[ ω ( E )] T ω ( E ) is a property of the system , not of the environment • A new (more restricted) derivation of the second principle • Interpretation of the constant κ : the average kinetic energy of a monoatomic gas at the temperature T is given by 3 κT , yielding κ = R/ (2 N A ) = 6 . 5 · 10 − 24 J / K ( k B = 2 κ ≃ 1 . 3 · 10 − 23 J / K ) 15
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