What is temperature? Peter Hänggi Universität Augsburg In collaboration with Dr. Gerhard Schmid Temperature sensation • Subjective temperature sensation • Memory effects • Measuring the heat flow • Heat – substance ↔ CldCold – substance What is temperature? Early temperature measurements No scale for measuring temperature between some reference points a thermoscope 1592: first documented (gas‐) thermometer invented by Galileo Galilei Galileo Galilei (1564 – 1642) First thermometer & temperature scales 1638: Robert Fludd – air thermometer &scale ~1700: linseed oil thermometer by Newton 1701: red wine as temperature indicator by Rømer 1702: Guillaume Amontons: Absolute zero temperature? 1714: mercury and alcohol thermometer by Fahrenheit DfiitiDefinition of tttemperature scales Olaf Christensen René Antoine Sir Isaac Newton Daniel Gabriel AdAnders CliCelsius Römer Ferchault de (1643 – 1727) Fahrenheit (1701 – 1744) (1644 – 1710) (1686 – 1736) Réaumur (1683 – 1757) Absolute Zero Ideal gas law: p V = NkB T V V =0,T=0 1848: Kelvin postulates an absolute zero temperature William Thomson ― Lord Kelvin T (1824 – 1907) 273.15 ◦C − It is impossible by any procedure to reduce the temperature of a system to zero in a finite number of operations. Absolute temperature scale Kelvin –scale (SI) Reference: absolute zero temperature & triple point of water (273.16K; 611.73 Pa) Scale division: 1K is equal to the fraction (273.16)‐1 of the thermodynamic temperature of the triple point of water ∆T =1K 1˚C (degree Celsius) ≡ How to determine the absolute thdhermodynamic temperature t dv T dt dt log = p dt T0 t0 ¡ ¢ Z v + cp0 dp ³ ´ t : arbitrary temperature scale v :volume cp0 : specific heat at constant pressure “ ... wo nun wieder unter dem Integralzeichen lauter direkt und verhältnismäßig bequem meßbare Größen stehen. ...” M. Planck Low temperature milestones Heike Kamerlingh 1908: liquid helium; 5 K Onnes (1853 – 1926) 1995: Bose‐Einstein‐ Condensate; 20 nK Eric A. Cornell Wolfgang Ketterle Carl E. Wieman 2003: BEC; 450 pK – W. Ketterle Temperature Extrema p + n ‐> quark gluon plasma with gold ion collisions in Relativistic Heavy Ion Collider (RHIC) 4 x 1012 °C Planck units Constant Symbol Value in SI units Speed of light c 299 792 458 m s‐1 Gravitational constant G 6.67429 ∙ 10‐11 m3 kg‐1 s‐2 reduced Planck’s constant¯h 1.054571628 ∙10‐34 J s 1 3 ‐2 ‐2 Coulomb force constant (4π²0)− 8 987 551 787.368 kg m s C ‐23 ‐1 BltBoltzmann constttant kB 1. 3806504∙10 J K Name Expression SI equivalents 5 1 2 32 Planck temperature TP = √¯hc G− k− 1.41168∙ 10 K Planck length3 1.61625∙ 10‐35 m lP = √¯hGc− 1 ‐8 Planck massmP = √¯hcG− 2.17644∙ 10 kg 5 ‐44 Planck timetP = √¯hGc− 5.39124∙ 10 s Planck units Name Expression SI equivalents 5 1 2 32 Planck temperature TP = √¯hc G− k− 1.41168∙ 10 K Planck length3 1.61625∙ 10‐35 m lP = √¯hGc− 1 ‐8 Planck massmP = √¯hcG− 2.17644∙ 10 kg 5 ‐44 Planck time tP = √¯hGc− 5. 39124∙ 10 s … ihre Bedeutung für alle Zeiten und für alle, auch außerirdische und außermenschliche Kulturen notwendig behalten unnd welche daher als natürliche Maßeinheiten bezeichnet werden können … … These necessarily retain their meaning for all times and for all civilizations, even extraterrestrial and non‐human ones, and can thhferefore be ddiesignated as natural units … (Planck, 1899) The highest temperature you can see Lightning: 30 000 °C Fuse soil or sand into glas Thermometers Gas Thermometers Idea l gas: V T = p Nk µ B ¶ Galilei Thermometer Volume change of water Upon increasing T → 4°C Water volume shrinks Linneaus thermometer Carl von Linné (1707 – 1778) RdthClilReversed the Celsius scale 1744: broken on delivery 1745: botanical garden in Uppsala Anders Celsius Noise Thermometer Johnson – Nyquist noise r oo V PSDV (ω)=2kBT R Resist -- classical regime only -- hmic OO PSDV : power spectral density of the voltage signal kB : Boltzmann constant R : resistance Quantum fluctuation‐dissipation theorem ¯hω PSD (ω) =¯hω coth ReY (ω) I 2k T µ B ¶ ¯hω ¯hω = 2 + ReY (ω) 2 exp(β¯hω) 1 µ − ¶ k T ¯hω B kBT ¯hω À ¿ P SDI (ω) = 2kBT ReY (ω) P SDI (ω) = ¯hω ReY (ω) Black body radiation 8πhc 1 Planck’ s law [1901]: u(λ, T ) = λ5 exp( hc ) 1 λ kBT − u(λ,T) : spectral energy density λ : wavelength h : Planck constant c : speed of light kB : Boltzmann constant Stefan – Boltzmann law: E T 4 ∝ Thermometer ! Small system – single realizations 1-st.1‐st law: law: ∆Eω = Qω + Wω ! random quantities ¡ 4 4 ... ∆U = Q + W h i ⇒ 4 4 Note: Q = ∆U W ∆U =? & W =? 4 − 4 4 weak coupling: Q = Qbath 4 − 4 REV Q 2‐nd law: ∆SA B 4 → ≥ T IRREV REV Q ∆SA B = 4 + ( 0) → T ≥ ∆U XW IRREV = − 4 + T Q IRREVX = − 4 bath + ; for weak coupling T X Cosmic background temperature T = 2.725 ±…. K Four Grand Laws of Thermodynamics Zeroth Law Transitivity ! A in equilibrium with B: fAB(pA, VA; pB, VB, ...) = 0 B in equilibrium with C: fBC(pB,VB; pC,VC,...)=0 A in equilibrium with C f (p , V ; p , V ,...) = 0 ⇒ ⇔ AC A A C C Allows the formal introduction of a temperature: T = TA(pA,VA; ...)=TB(pB,VB; ...)=TC(pC,VC; ...) First Law Julius Robert James Prescott Hermann von von Mayer Joule Helmholtz (1814 – 1878) (1818 – 1889) (1821 – 1894) Firs t Law – Energy CtiConservation ∆U = Q + W ∆U change in internal energy Q heat added on the system 4 W work done on thesystem 4 H. von Helmholtz: “Über die Erhaltung der Kraft” (1847) ∆U = (T ∆S)quasi-static (p∆V )quasi-static − Second Law Rudlfdolf Julius Emanuel Clilausius William Thomson alias Lord Kelvin (1822 – 1888) (1824 – 1907) Heat generally cannot No cyclic process exists whose sole spontaneously flow from a effect is to extract heat from a material at lower temperature to single heat bath at temperature T a materilial at hig her temperature. and convert it entilirely to work. δQ = T dS (Z¨urich, 1865) Theeodyarmodynamic Teepeauemperature δQrev = TdS thermodynamic entropy ← S = S(E,V,N1,N2, ...; M,P,...) S(E,...): (continuous) & differentiableand monotonic function of the internal energy E ∂S 1 = ∂E T µ ¶... Entropy S – content of transformation „Vdlt“Verwandlungswert“ d S = δQrev T ; δQirrev < δQrev V2,T2 δQ Γrev Γirrev 0 C T ≤ I 1 C = Γrev− + Γirrev V1,T1 δQ S(V ,T ) S(V ,T ) 2 2 − 1 1 ≥ T ∂S ZΓirrev δQ 0 NO ! S(V ,T ) S(V ,T )= 2 2 − 1 1 T ∂t ≥ ZΓrev Perpetuum mobile of the second kind ? One heat reservoir Two heat reservoirs NO ! Third Law Approaching zero temperature Famous exception of the 3rd law … and Planck’s version THIRD LAW AND PLANCK 1913 Beim Nullpunkt der absoluten Temperatur besitzt die Entropie eines jeden chemisch homogenen festen oder flüssigen Körpers den Wert Null. ∂E ¯hω ¯hω Specific heat: CE = , E = 0 + 0 ∂T 2 exp (¯hω0β) 1 − Quantum harmonic Oscillator: the limit ω 0 does NOT yield the specificheatCE = k /2 0 → B of the free particle 1 NiilthNo virial theorem ! E = k T No equipartition th.! 6 2 B Specific Heat E = E exp( βE ) exp( βE ) i − i − i i i X ± X E dE 2 1 2 C = =(k T )− (E E ) > 0 ! dT B h i − h ii i ∂S ∂2S 1 Note also: =1/T , = - ∂E ∂E2 CET 2 S CE<0 CE>0 E Temperature Fluctuations – An Oxymoron Ch. Kittel, Physics Today, May 1988, p. 93 1 Nano‐system β = BATH kBT T does NOT fluctuate! ∆U ∆β =1 ∆U ∆T = k T 2 NO! | || | ↔ | || | B “…Tempppyyyerature is precisely defined only for a system in thermal equilibrium with a heat bath: The temperature of a system A, however small, is defined as equal to the temperature of a very large heat reservoir B with which the system is in equilibrium and in thermal contact. Thermal contact means that A and B can exchange energy, although insulated from the outer World. …” Molecular Dynamics • Every quadratic term in the Hamiltonian 1 contributes kB T to the internal energy 2 • Classical systems: VIRIAL THEOREM n n [Hi q or p Ei = kBT/n] ∝ i i ⇒ h i • Classical harmonic Oscillator p2 1 1 h i i = m v2 = k T 2m 2 h i i 2 B 1 1 mω2 x2 = k T 2 h i i 2 B NOT SO FOR A QUANTUM OSCILLATOR Specific heat paradox (microcanonical) R. Em den (1907); A. S. Edding ton (1926); D. Lynden-Bell & R. Wood (1968); W. Thirring (1970); W. Thirring, H. Narnhofer & H. A. Posch (2003) Coulomb: Φi = kBT CLASSICAL ! h i − 1 Tkin = Φtot =3NkBT/2 −2h i 2Tkin + Φtot =0=E + Tkin h i dE dT CE = = kin = 3Nk /2 < 0 dT − dT − B Relativistic Regime Σ(u = 0) Σ0(u = 0) −→ 6 x0 = γ(u) (x u t) − · 2 t0 = γ(u) t ux/c − 1/2 ¡ u2 −¢ with γ(u)= 1 − c2 µ ¶ 2 2 2 2 2 2 x c t = (x0) c (t0) − − Note: thermodynamic observables are NONLOCAL J. Dunkel & P.H., Relativistic Brownian motion, Phys. Rep. 471, 1 - 73 (2009) Maxwell‐Boltzmann distribution velocity distribution: m 3/2 v2 + v2 + v2 f (v ,v ,v )= exp x y z ~v x y z 2π k T − 2 k T/m µ B ¶ à B ! 25 °C James Clerk Maxwell Ludwig Eduard Boltzmann (1831 – 1879) (1844 – 1906) ! v < c violated ! Diffus ion classical relativistic Author's personal copy Author's personal copy J. Dunkel, P. Hänggi / Physics Reports 471 (2009) 1–73 5 J. Dunkel, P. Hänggi / Physics Reports 471 (2009) 1–73 45 that point on, nonrelativistic statistical mechanics emerges without much difficulty [287,288].