What Is Temperature?

Home , Anders Celsius, Guillaume Amontons

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 CClielsius 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 didesignated 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 − ¶

kBT ¯hω k T ¯hω À B ¿

PSDI(ω) = 2kBT ReY (ω) PSDI(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 = Q 4 − 4 bath REV Q 2‐nd law: ∆SA B 4 → ≥ T IRREV REV Q ∆SA B = 4 + ( 0) → T ≥ IRREV ∆U XW = − 4 + T IRREVX Q = − 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 CCtionservation ∆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) (p∆V ) quasi-static− quasi-static Second Law

Rudlfdolf Julius Emanuel Clliausius 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 Γ 0 irrev T ≤ IC 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 (¯hω β) 1 0 − Quantum harmonic Oscillator:

E the limit ω0 0 does NOT yield the speciﬁcheatC = kB/2 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 [H qn or pn E = k T/n] i ∝ i i ⇒ h ii B • 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: Φ = k T CLASSICAL ! h ii − B 1 T = Φ =3Nk T/2 kin −2h toti B 2T + Φ =0=E + T kin h toti kin 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 ¶ Ã 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]. Unfortunately, the situation 5. Non-Markovianbecomes diffusion significantly processes more in Minkowski complicated spacetime in the relativistic case: Due to their finite propagation speed, relativistic interactions should be modeled by means of fields that can exchange energy with the particles [6]. These fields add an infinite number The preceding section has focused on relativistic Brownian motions in phase space. In the remainder we will discuss relativistic diffusionof degrees models of in freedom Minkowski to spacetime, the particle i.e., continuous system. relativistic Eliminating stochastic the field processes variables that do from not explicitly the dynamical equations may be possible depend on thein momentum some cases, coordinate. but this On procedure the one hand, typically such spacetime leads processes to retardation may be constructed, effects, i.e., for the example, particles’ from equations of motions become non- a Brownian motionlocal in process time [in220 phase,221 space,244 by,245 integrating,249,250 out]. theThus, momentum in special coordinates. relativity As it a isresult usually of this very averaging difficult or even impossible to develop a procedure, the reduced process for the position coordinate will be non-Markovian. Alternatively, one can try to derive or postulate a relativisticconsistent diffusion field-free equation Hamilton and/or diffusion formalism propagators of interacting in spacetime particles. on the basis of microscopic models [10, 28,153,366] or plausibilityIn spite considerations of the difficulties [23]. Regardless impeding of the a approach rigorous adopted, treatment in order of to classical comply with relativistic the principles many-particle systems, considerable of special relativity,progress the was resulting made spacetime during processthe second must halfbe non-Markovian, of the past century in accordance in constructing with the results an ofapproximate Dudley relativistic kinetic theory [167, (Theorem 11.3225 in [,255333]),289–309 and Hakim] (Proposition based on 2 relativistic in [338]). Roughly Boltzmann speaking, equations this means that for any the relativisticallyone-particle acceptablephase space probability density functions generalization of the classical9 diffusion equation (1) should be of at least second order in the time coordinate. The construction(PDFs). andFrom analysis such of arelativistic kinetic theory,diffusion itmodels is only in Minkowski a relatively spacetime small step poses to an formulating interesting problem a theory of relativistic Brownian motion in its own right.processes Additionally, in terms the investigation of Fokker–Planck of these processes equations becomes and relevant Langevin in view equations. of potential While analogies the with relativistic Boltzmann equation [311, relativistic quantum312] is theory a nonlinear [383,391],partial similar to integro-differential the analogy between Schrödinger’s equation for equation the PDF, and the Fokker–Planck diffusion equation equations(1) are linear partial differential in the nonrelativisticequations case and, [501,502 therefore,]. The present can section be more intends easily to provide solved an or overview analyzed over [classical73]. relativistic diffusion models that have been discussed in the literature [10,23,27–29,337,366,367,379–381,391]. For this purpose, we first recall basic propertiesThe of the present Wiener (Gaussian) article focuses process, which primarily constitutes on the relativistic standard paradigm stochastic for nonrelativistic processes diffusions that are characterized by linear evolution in position spaceequations (Section 5.1 for).their Subsequently, respective relativistic one-particle generalizations (transition) of the nonrelativistic PDFs. The diffusion corresponding equation (1) and/or phenomenological theory of relativistic the nonrelativisticBrownian Gaussian motion diffusion and propagator diffusion will processes be discussedAuthor's has [23]. experienced personal considerable copy progress during the past decade, with applications in various areas of high-energy physics [315–322] and astrophysics [323–327]. From a general perspective, relativistic 5.1. Reminder: Nonrelativistic diffusion equation stochastic processes provide a useful approach whenever one has to model the quasi-random behavior of relativistic We start byparticles briefly summarizing in a complex a few environment. relevant facts aboutJ. Dunkel, Therefore, the P. standard Hänggi it / may nonrelativistic Physics be Reports expected diffusion 471 (2009) that equation 1–73 relativistic [287,339 Brownian,422, motion and diffusion49 concepts 502] will play an increasingly important role in future investigations of, e.g., thermalization and relaxation processes in ∂ Similarly, the singular diffusion fronts predicted by Eq. (209) represent a source of concern if one wishes to adopt the astrophysics2 [323–326] or high-energy collision experiments [315,316,318,319,328,329]. " D ", t t0, (194) ∂telegrapht = ∇ equation≥ (206) as a model for particle transport in a random medium. While these singularities may be acceptable where Din the> 0 case denotes of photon the spatial diffusion diffusion [371–375 constant,], and they"( seemt, x) unrealistic0 the one-particle for massive PDF for particles, the particle because positions such fronts would imply that d 1.2. Relativistic diffusion processes: Problems and≥ general strategies x R at time t. Within classical diffusion theory, Eq. (194) is postulated to describe the (overdamped) random2 motion ∈ a finite fraction of particles carries a huge amount of kinetic energy (much larger than mc ). In view of these shortcomings, it of a representativeappears reasonable particle in a to fluctuating explore otherenvironment constructions (heat bath). of In relativistic particular, Eq. diffusion(194) refers processes to the rest [23 frame,390]. of In the the next part we will discuss bath. According to our knowledge, the first detailed mathematical studies on relativistic diffusion processes were performed a different approach [23] that may provide a viable alternative to the solutions of the telegraph equation. There existindependently several well-known by waysŁopusza to motivatenski` [ or330 derive], Rudberg the phenomenological [331], and diffusion Schay [ equation332] between(194) by means1953 of and 1961. In the 1960s and 70s their microscopicpioneering models (see, e.g., work [287, was339,422 further,502]). With elaborated regard to byour subsequentDudley who discussion published of relativistic a series alternatives, of papers it is [333–336] that aimed at providing useful to5.3. briefly Relativistic consider a diffusion ‘hydrodynamic’ propagator derivation [503], which starts from the continuity equation ∂ an axiomatic approach to Lorentz invariant Markov processes [74] in phase space. Independently, a similar program was "(tpursued, x) byj(t Hakim, x), [220–222,337,338], whose insightful analysis helped to elucidate(195) the conceptual subtleties of relativistic ∂t In principle,= −∇ · one can distinguish two different routes towards constructing relativistic diffusion processes: One can either where jtry(t, x to)stochasticdenotes find an the acceptable processescurrent density relativistic [338 vector.]. Dudley In diffusion order (Theorem obtain equation, a closed 11.3 or equation inone [333 can for]) focus the and density directly Hakim", the (Proposition on current the structurej has to 2 be in of [ the338 diffusion]) proved propagator. the non-existence of 10 expressedIn in thenontrivial terms present of ". One partLorentz way we of doingshall invariant this consider is to Markov postulate the latterprocesses the following approach in rather Minkowski [23 general]. The ansatz basic spacetime, {cf. idea Eq. is (2.81) asto alreadyrewrite in [503]} the suggested nonrelativistic by Łopusza diffusionnski` [330]. This propagatorfundamental(199)t in result such impliesa form that that its it relativistic is difficult generalization to find acceptable follows relativistic in a straightforward generalizations manner. of the This well-known can be achieved nonrelativistic j(t, x) dt% K(t t%) "(t%, x), (196) be re-expressingdiffusion= −∇ equation Eq.−(199) [287in, terms339] of an integral-over-actions. !t0 For this purpose, we consider a nonrelativistic particle traveling from the event x0 (t0, x0) to x (t, x) and assume where, in general, K may∂ be a memory kernel. However, considering for the moment the memory-less kernel¯ function= 74 ¯ = that the particle" canD experience2", multiple scatterings on its way, and that the velocity is approximately constant between (1) KF(t t%) 2∂D δ(t= t%),∇ (197) two− successive:= t scattering− events. Then the total action (per mass) required along the path is given by one finds d Diffusion:where D > space-time0 ist the diffusion constantonly and "(t, ) 0 the PDF for the particle positions at time t. In order to 1 x 194 J Masoliver andGHWeiss x R j (t, x) D "(t, x). 2 ≥ (198) ∈ F circumvent=a(−x x0∇) this ‘no-go’dt! v(t! theorem) , for relativistic Markov processes in spacetime, one usually adopts either of(213) the following ¯|¯ = 2 11 on the boundary. Likewise the reflecting boundary Upon insertingtwo this strategies expression! intot0: the continuity equation (195), we recover the classical diffusion equation (194). condition requires that the normal component of the flux Now, it has been well-known for a long time that the diffusion equation (194) is in conflict with the postulates of specialinto the boundary should be equal to zero. Because where the velocity v(t!) is a piecewise constant function, satisfying the property of persistence inherent in the telegrapher’s One considers non-Markovian diffusion processes (t) in Minkowski2 spacetime∂ 2/∂ 2 [10equation,23 is, analogous365–367 to the physical]. property of relativity.classical To briefly diffusion illustrate this, (Markovian we specialize to) simplest case of d 1 spaceX dimensions, where x . In thismomentum it is necessary to take into account the • t = ∇ = case, the propagator of Eq. (194) at times t > t0 is given by the Gaussian direction in which the particle is travelling in deriving One constructs relativistically acceptable Markov processes in phase space byboundary considering conditions. The analysis directed not at findingonly an the position •x x0 dt! v(t1!/).2 exact form of the resulting boundary conditions in(214) one coordinate X(t) of the diffusing particle,2 but also its momentum coordinate P(t) [11–22dimension is based,24 on,26 an examination,31,32 of, the220–222 functions ,332–338]. = + t 1 (x x0) ( , , ) 0 − . an(j) and bn(j) in equations (13) and (14). p t x t0 x0 ! exp (199)To determine the form of the boundary condition it | = 4π D(t t0) − 4D(t t0) is sufficient to consider a single point j 0 and a Clearly, the" nonrelativistic# action" (213) becomes# minimal for the deterministic (direct) path, i.e., if the particle= does not − − persistent random walk on the half-line j ! 0, thereafter The propagator1.2.1.(199) Non-Markovianrepresents the solution diffusion of Eq. (194) modelsfor thein Minkowski initial condition spacetime passing to the continuum limit. We can derive several collide at all. In this case, it moves with constant velocity v(t!) (x x0)/(t t0) for all t! typest of0, boundaryt , yielding conditions by assuming the that smallest when a "( , ) δ( ). ≡ − − ∈random[ walker] reaches j 0 it is either trapped there telegraphpossiblet0 x A actionequation commonlyx x0 value considered(non-Markovian ‘relativistic’) generalization of Eq. (1) is the telegraph equation [10,365–367= ] =12 − with probability θ, provided that it is moving towards That is, if X(t) denotes the random path of a particle with fixed initial position X(t ) x , then p(t, x t , x )dx givesthe exterior of the line segment, or else it is reflected 2 0 0 0 0 back to the point from which it came with probability (x x0) = | 1 θ. The case of θ 0 or 1 corresponds to what ( ) 2 . is− termed in chemical physics%= the radiation boundary a x x0 ∂ −∂ condition. Suppose that the particle is reflected back(215) to 74 − 2 t δ( ) ( ) ( )/ . The factor ‘2’ in Eq. (197)¯τ|¯v appears="2 because(t t of"0 the) conventionD "t, dt% t t% f t% f t 2 j 1 at step n 1. At this step it must have come (2) 2 − 0 − = from= j 0, and+ at the immediately preceding step it ∂t + ∂t = ∇ must have= been at j 1. Hence we have the boundary On the other hand, to match the boundary$ conditions it is merely required that the mean velocity equals= (x x )/(t t ). condition − 0 − 0 an 1(1) (1 θ)bn(0) (38) Consequently, in the nonrelativistic case, the absolute velocity of a particle may become arbitrarily+ = − large during some or, in the continuum limit a(0, t) (1 θ)b(0, t). alternativeintermediateaimed atapproach constructing time interval a relativistict!, t!! quantumt0, t theory. Hence, for the interacting largest many-particle possible action systems value [250–253 is a (x]., Forx0) aHowever, detailed it is. discussionnot These this boundary considerations of= condition relativistic− that is many-particle Figure 1. The development of the one-dimensional+ interesting but rather the boundary condition to be theory, we refer to the insightful[ considerations] ⊂[ ] in the original papers of Vansolution Dam to the telegrapher’s and Wigner equation as [249 a function,¯250 of¯ ] and=+ Hakim∞ [220–222,224] as well as to the recent put us in the position to rewrite the Wiener propagator (199) as an integral-over-actions: imposed on the function p(x, t x0). At the boundary time, showing the evolution from wave-like behaviour to equation (38) implies that | review by Hakim and Sivak [286]. diffusive behaviourJ Masoliver as well as& theG deltaH Weiss functions at the a (x x ) extremes ofEurp(x ,Jt) .Phys The delta 17 function:190 (1996) contributions at p(0, t) a(0, t) b(0, t) (2 θ)b(0, t). (39) 9 0 the endpoints are not included in the figure. = + = − Comprehensive+ introductions¯|¯ to relativistica Boltzmann equations can be found in the textbooks by StewartWe also have [310 the relation], de Groot et al. [311], and Cercignani p(x x0) da exp , θ (216a) and Kremer [312], or also in the reviews by Ehlers [313] and Andréasson [that314 the]. appropriate solution is a constant, µ (t) x . a(0, t) b(0, t) θb(0, t) p(0, t). (40) ¯|¯ ∝ a (x x ) − 2D 1 = 0 − = = 2 θ 10 0 In a similar fashion we find that the second moment is − A diffusion! process− ¯|¯ is considered" as ‘nontrivial’# if a typical path has a non-constant,the solution to the equation non-vanishing velocity.We finally return to equations (17) and (18), subtracting the latter from the former. In this way we supplemented by the normalization condition d2µ 1 dµ 11 2 2 2v2 (36) find that a b is related to the total density by The mathematical interest in relativistic diffusion processes increasedd int 2 + theT dt 1980s= and 1990s, when several− authors [340–362] considered the ∂ ∂p 1 PRD 75:043001 (2007) subject to the initial conditions µ (0) x2 and (a b) v (a b), (41) possibility of extending Nelson’s stochastic quantization approach [363] to the framework of special2 = 0 relativity.∂t These− = − studies,∂x − T − although interesting from a dµ2/dt t 0 0. Again this simple differential equation mathematical1 dxp point(x x of0). view, appear to have relatively little physical relevanceis readily| = solved,= because the solution Nelson’s being stochasticwhich dynamics is to be evaluated [363 at x ] fails0. The to combination reproduce(216b) the correct of equations (40) and (41) yields= a single (radiation) = ¯|¯ 2 2 t/T µ (t) x 2v T [t T(1 e− )]. (37) quantum correlation functions even in the nonrelativisticRelativistic case thermodynamics [364]. & BM 2 = 0 + − − boundary condition, [7], for the function p(x, t x0): ! 2 2 2 | When t T this becomes µ2(t) x0 v t consistent ∂p θ ∂ 1 The representation12 (216) may be generalized to the relativistic casewith in a a" wave straightforward propagating at uniform≈ + speed. manner: In the Onev merely needsp . to insert(42) Masoliver and Weiss [10] discuss several possibilities of deriving Eq. (2) from different underlying2 models.∂x x 0 = 2 θ ∂t + T x 0 opposite limit, t T , µ2(t) 2v Tt which is the ! = − " # ! = $ ≈ ! ! the corresponding relativistic expressions into the boundaries ofresult the obtained integral from a diffusion(216a) process. . A commonlyThe two most! commonconsidered boundary conditions! relativistic correspond to absorption! and reflection. The boundary! condition in generalization of Eq. (213), based on the particle’s proper time, reads [6] the case of trapping requires setting θ equal to unity. 4. Boundary conditions The resulting boundary condition is quite unlike that t in the case of the diffusion equation, since it includes 1/2 It is well known that the boundary condition for the derivatives with respect to both t and x. Parenthetically 2 diffusion equation which corresponds to an absorbing we note that the trapping boundary condition is also a(x x0) dt! 1 v(t!) . boundary is p(!, t) 0, where ! consists of all points equivalent to setting a(0, t) 0 which is understandable(217) ¯|¯ = − − = = !t0 Analogous to the nonrelativistic$ case,% the relativistic action (217) assumes its minimum a for the deterministic (direct) − path from x to x, characterized by a constant velocity v(t ) (x x )/(t t ). One explicitly obtains 0 ! ≡ − 0 − 0 2 2 1/2 a (x x0) (t t0) (x x0) , (218a) − ¯|¯ = − − − − i.e., a is the negative Minkowski distance of the two spacetime events x0 and x. The maximum action value a 0 is − $ % 79 ¯ ¯ + realized for particles that move at light speed. Hence, the transition PDF for the relativistic generalization of the Wiener= process reads

1 a (x, x0) p(x x ) N − exp − ¯ ¯ 1 , (218b) ¯|¯0 = − D − & ' 2 ( ) 2 2 if (x x0) (t t0) , and p(x x0) 0 otherwise, with a determined by Eq. (218a). − ≤ − ¯|¯ ≡ −

79 In general, particles must undergo reflections in order to match the spatial boundary conditions. Jüttner Gas f (p~)=[β/(2πm)]d/2 exp βp2/2m Maxwell − 1 2 4 2 2 1/2 f (p~) = Z− exp β (m c¡ + p c ) ¢ J¨uttner d − J u =0 h i

p~ ~v = dk = d/β h · i BT J u =0 statistical realtivistic temperature 1 T = =(k β )− T B J

J. Dunkel & P.H., Phys. Rep. 471, 1-73 (2009) moving bodies appear cooler

Relativistic TD & BM DPG, March 2010 Relativistic TD & BM DPG, March 2010 moving bodies appear hotter

16 Heinrich Ott (1894-1962) photo provided by Oscar Bandtlow (QMUL)

17 Relativistic TD & BM DPG, March 2010 “Temperature” problem in RTD? 1923/1963 .. hotter!

1907/08 moving bodies appear cooler

T=T’ 1966-69

1940s maybe ... not CK Yuen, Amer. J. Phys. 38:246 (1970)

Relativistic TD & BM DPG, March 2010

Time parameters and (relative) entropy

EPL 87: 30005 (2009)

Relativistic TD & BM DPG, March 2010 ? Negative Temperature ?

Spin system: S~ =1/2 ; μ~ = γS~ ; H = μ~ B~ | | − i · X 1 1 S~ B~ Two-State-System: ² = γB < ² = + γB = μB k ⇒ g −2 e 2

N = n + n & E = μB(n n ) , typically E<0 g e e − g N! 1 E Ω = S = kB ln Ω n = N ng! ne! ⇒ g 2 − μB µ ¶ 1 E 1 ∂S n = N + e 2 μB = µ ¶ ⇒ T ∂E Negative (Spin)‐Temperature! Take Home Messages absolute temperature T 0 • ∃ ≥ S T = ∂E • ∂S ... with S(E) a concave function of E ! E S(E) : microcanonical NOT always = S(E) : canonical • ( long range, non-ergodic parts )

T does NOT ﬂuctuate • quantum mechanics: no virial theorem; no equipartion • relativistic statistical thermometer • = TΣ TΣ0

Jüttner Gas f (p~)=[β/(2πm)]d/2 exp βp2/2m Maxwell − 1 2 4 2 2 1/2 f (p~) = Z− exp β (m c¡ + p c ) ¢ J¨uttner d − J u =0 h i

p~ ~v = dk = d/β h · i BT J u =0 statistical relativistic temperature 1 T = = (k β )− T B J

J. Dunkel & P.H., Phys. Rep. 471, 1-73 (2009) moving bodies appear cooler

Relativistic TD & BM DPG, March 2010 Comparision of temperature scales Kelvin [K] Celsius [°C] Fahrenheit [°F] Absolute zero 0.00 ‐273.15 ‐459.67 Fahrenheit’s ice/salt 255.37 ‐17.78 0.00 mixture Ice melts (at (100) = 273.15 32.00 standard pressure**) 0.00 * Average human 309.95 36.80 98.24 body temperature Water boils (at 373.13 (0) = 100 * 211.97 standard pressure) Titanium melts 1941.00 1668.00 3034.00 source: wikipedia.org Further temperature scales: Newton, Delisle, Réaumur, Rømer, M. Planck (linear scales) Rudolf Plank (logarithmic scale) * Celsius used a reversed temperature scale with “0” indicating the boiling point of water and “100” the melting point of ice ** 1 atm = 760 mmHg = 101325 Pa

Temperature definitions Overview

I. History

II. Temperature scales

III. Thermometers

IV. GdGrand Laws of Thermod ynami cs

V. TtTemperature chtitiharacteristics Temperature scales

Temperatur: Was ist das eigentlich ?

Peter Hänggi Universität Augsburg

In Zusammenarbeit mit Dr. Gerhard Schmid What is temperature?

Peter Hänggi Universität Augsburg

In collaboration with Dr. Gerhard Schmid Planck units

Internal energy: E(S, V, N )=TS pV + μ N { i} − i i X Helmholtz free energy: F (T,V, N )=E TS { i} − Enthalpy: H(S, p, N )=E + pV { i} Gibbs free energy: G(T,p, N )=E + pV TS= μ N { i} − i i X Planck potential: Φ(T,p, N )= G/T { i} − Quantum Regime

34 4 h =6.62606896 10− Js= 2 · KJ RK 2 with KJ =2e/h , RK = h/e

hν =1kgc2 ν = 135639273 1042Hz → · Quantum Harmonic Oscillator

2 p M 2 2 1 1 HS = + ω0x ,En =(n + )¯hω0 , β = 2M 2 2 kBT Relativistic Brownian motion

J. Dunkel, P.H., Phys. Rep. 471, 1 (2009)

1 +1 l0(u)=l0γ− t0(u)=t0γ

α T 0(u)=T0γ−

Masoliver & Weiss, Eur. J. Phys. 17: 190 (1996) Relativistic thermodynamics Moving Observers Σ(u =0) Σ0(u =0) −→ 6

3 1 mγ(v0) 2 f 0 (v0,u)=Z− exp β γ(u)mγ(v0)(c + uv0) J¨uttner 1 γ(u) − J

v0 : velocity in mov£ing frame ¤

u =0.2c uu =0=0..22cc

J. Dunkel & P.H., Phys. Rep. 471, 1-73 (2009) Measuring temperature in Lorentz invariant way Lorentz invariant equipartition theorem 3 2 k = mγ(u) γ(v0)(v0 + u) BT h it0 with u = v0 −h it0

u/c u/c A statistical definition of Temperature

Ω(E, V, N) :numberofmicrostatesforagiven macrostate(E, V, N)

E1,V1,N1 E2,V2,N2 E1, V1, N1 E2, V2, N2 Ω (E ,V ,N ) Ω (E ,V ,N ) Ω = Ω Ω 1 1 1 1 2 2 2 2 ⇒ 1 · 2 ∂Ω1 ∂Ω2 in equilibrium: δΩ = δE1 Ω2 + Ω1 δE2 =0 ∂E1 ∂E2 1 ∂Ω1 1 ∂Ω2 energy conservation: δE1 = δE2 = − ⇒ Ω1 ∂E1 Ω2 ∂E2 Boltzmann (Planck): S = kBlnΩ 1 1 = T1 = T2 = T ⇒T1 T2 ⇒ Typical temperature values [°C]

Boiling point of Nitrogen ‐195.79

Lowest recorded surface temperature on Earth ‐89 (Vostok, Antarctica –July 21, 1983)

Highest recorded surface temperature on Earth 58 (Al’ Aziziyah, Libya – September 13, 1922) Temperature in the Earth’s Thermosphere ~ 1500 (80 ‐ 650 km above the surface)

Melting point of diamond 3547

Surface temperature of the sun (photosphere) ~ 5526

Temperature in the interior of the sun ~ 15∙106 Brief history of the temperature concept