Temperature

In General > s.a. heat; thermodynamics; units.
* Idea: The parameter governing the thermal equilibrium between one part of an isolated system and another; Can be defined in general as the rate of energy increase per unit increase in the state uncertainty under no-work conditions; Or, à la Carathéodory, temperature is the "right" integrating factor of the exchanged heat between the system and a heat bath.
* History: Daniel Fahrenheit (1686–1736) invented a mercury thermometer capable of reproducible measurements; Joseph Black (ca 1760) made possible the transition from thermoscopes, which register qualitative differences, and thermometers.
\$ Def: The intensive variable thermodynamically conjugate to energy,

T −1:= ∂S(E,V) / ∂E ,    or   T = ∂U / ∂S|V .

* In statistical mechanics: It can be defined for a microcanonical ensemble as T −1:= ∂S(E,V)/∂E; For a canonical ensemble it is then the temperature of the microcanonical ensemble composed of the system + heat bath [@ but see Mandelbrot PT(89)jan for a claim that only its fluctuation can be defined in a unique way for a microcanonical ensemble].
* Lowest values: 1960s, The lowest Ts attained are ≤ 10−6 K, with magnetic cooling (Precool down to about 1 K with liquid He, apply a magnetic field which aligns the atoms while in contact with the bath, remove contact with bath, then switch field off; The performance of magnetic work by the thermally isolated system of spins, and using the first law of thermodynamics, cools the system further); 2003, Bose condensate of sodium atoms cooled down to ≤ 0.5 × 10−9 K; 2014, Plans to reach 100 pK in the Cold Arom Lab on the ISS.
* Highest values: 2006, T = 3.6 × 109 K measured at Sandia National Lab; 2010, T = 4 × 1012 K measured in the quark-gluon soup produced at Brookhaven Lab's Relativistic Heavy Ion Collider; 2012, The temperature at the LHC is 30% higher than the value achieved by RHIC, but official values have not yet been published; 2015, The highest melting point is that of a combination of hafnium, nitrogen and carbon, and is expected to be about 7,460 degrees Fahrenheit – about two-thirds the temperature of the sun [@ Hong & van de Walle PRB(15) + news WashP(15)jul].
* Locality: A subsystem of a large traditional thermal system is in a thermal state at the same temperature, but for strongly interacting systems the locality of temperature breaks down.
@ Lowest temperatures in experiments: news ej(12)may [–273.1497°C at the University of Alberta]; news sn(17)jul [update]; news sn(17)aug [molecules].
@ Highest temperatures in experiments: news livesci(06)mar; news disc(10)feb; news bnl(12)jun, lat(12)jun.

Specific Systems and Effects > s.a. black-hole thermodynamics; ising model [roughening T]; quantum fields in curved spacetime.
* Negative T: It can occur in quasi-equilibrium systems, if one starts with an equilibrium state of positive T and quickly changes the parameters so that higher-energy states are more populated, and that T −1:= ∂S/∂E changes sign; Or in systems such as spin systems, where the number of states available at high energies is low because all spins become aligned.
* Relativistic T, theory: The idea has been debated for a long time; Einstein and Planck thought, at one time, that a speeding thermometer would measure a lower temperature than one in the gas rest frame, while others thought the temperature would be higher; The only clear thing is that absolute zero is invariant; (Sewell) In both the special relativistic and non-relativistic settings, a state of a body cannot satisfy the KMS (Kubo, Martin and Schwinger) thermal equilibrium conditions for different inertial frames with non-zero relative velocity; In that sense, there is no law of temperature transformation under either Lorentz or Galilei boosts.
* Relativistic T, experimentally: 2007, Direct experimental results have not been obtained because of the difficulty in containing a gas moving at relativistic bulk velocities, but there is hope to get evidence from some astrophysical systems, and extensive simulations suggest that the temperature in a moving frame is the same as that measured in the rest frame.
* Non-equilibrium T: A definition has been proposed using the fluctuation-dissipation relation, but the value one obtains may depend on the observable.
@ Microcanonical T: Davis & Blakie JPA(05)cm [classical Bose gas].
@ Negative T: Lavenda JPA(99) [argument against]; news PhysOrg(13)jan, nat(13)jan [gas at negative temperature obtained]; Schneider et al a1407; Hama et al PRL(18); de Assis et al JPB(19)-a1805 [experimentally feasible platform]; Volovik a2104 [extension to relativistic field theories].
@ Relativistic T: Komar GRG(95); Costa & Matsas PLA(95)gq; Landsberg & Matsas PLA(96)phy, PhyA(04) [no relativistic transformation]; Cubero et al PRL(07)-a0705 + news pn(07)oct [simulations]; Wu EPL(09)-a0804 [inverse temperature 4-vector]; Rasinariu a0804 [moving systems appear cooler]; Sewell JPA(08)-a0808, RPMP(09), JPA(10)-a1010 [not well defined]; Mi et al MPLA(09) [and blackbody radiation]; Mitchell & Petrov EJP(09) [moving medium]; Gransee a1609 [quantum Klein-Gordon field in Minkowski space, spacetime dependence]; Hoshino & Nakamura a1807 [holographic approach]; Hao et al a2105 [in Minkowski and Rindler spacetimes]; > s.a. generalized thermodynamics; thermal radiation.
@ Quantum system: Mitchison et al a2103 [for a pure quantum state]; > s.a. Eigenstate Thermalization Hypothesis.
@ Non-equilibrium T: Essex et al AJP(03)oct [radiation]; Bertin et al PRL(04) [lattice, with conserved energy]; Carati PhyA(06); Martens et al PRL(09) [fluctuation-dissipation temperature]; Cugliandolo JPA(11) [from deviations from the equilibrium fluctuation-dissipation theorem]; Colombani et al PRL(11) [experiment].
@ In non-extensive statistical mechanics: Hansen NA(05)ap [pseudo-T for gravitating clusters]; Abe PhyA(06) [Tsallis entropy]; > s.a. statistical mechanics.
@ Small systems: Liu & Wang PLA(08) [finite number of classical spin-half particles]; Yan et al PhyA(09) [different definitions]; news sn(19)mar [nanoelectronics below 1 mK]; Vallejo et al a2005 [finite-dimensional quantum system].
@ Cold matter: Leanhardt et al Sci(03)sep + pw(03)sep [BEC at 500 pK]; Beige et al BJP(05)qp/04-proc [cooling N particles to very low T]; news bbc(09)jul [Planck observatory at 0.1 K]; Stamper-Kurn Phy(09); news sa(15)jun [molecules at 500 nK]; news sn(20)jan [nanoparticle in translational ground state]; Whittle et al a2102 [10-kg object at 77 nK]; > see condensed matter [supercooled liquids]; Lasers [laser cooling]; metamaterials [granular matter]; molecules [ultracold].

References > s.a. thermal radiation; units [definition of K].
@ General: Ehrlich AJP(81)jul [concept]; Beghian NCB(93); Rugh PRL(97) [dynamical approach]; Shachtman 99 [history; I]; Ferraro et al EPL(12)-a1102 [intensive nature of temperature and quantum correlations]; Biró 11; Skow PhSc(11) [metric structure]; Mares TMAC(15)-a1604 [relationship between temperature in statistical theory and phenomenological temperature]; Beretta & Zanchini AAPP(19)-a1911 [beyond equilibrium states of macroscopic systems].
@ Limitations of concept: news Nat(04)aug [meaningless for nanotubes]; Hartmann & Mahler EPL(05)cm/04 [spin-1 chain].
@ Local definitions: Hartmann CP(06)cm [minimum length scales]; García-Sáez et al PRA(09)-a0808 [quantum]; Kliesch et al PRX(14)-a1309 [spin and fermionic lattice systems]; Hernández-Santana et al NJP(15)-a1506 [interacting spin chains].
@ Measuring temperature: Weld et al PRL(09) + Rey Phy(09) [down to 50 pK, for ultracold atoms in optical lattices]; Stace PRA(10)-a1006 [quantum limits to precision thermometry]; Sherry SHPSA(11) [thermoscopes, thermometers and measurement]; Mann & Martín FP(14)-a1405 [using the Berry phase to construct a precision quantum thermometer]; news dm(14)jun [most sensitive]; news Phy(14)aug [using quantum dots to measure mK temperatures]; Jarzyna & Zwierz PRA(15)-a1412, Xie et al PLA(17)-a1608 [interferometric thermometer]; Mehboudi et al PRL(19) [in a Bose-Einstein condensate, sub-nK, using polarons]; Clark Phys(20)jan [extremely low T]; Ehnholm & Krusius a2010 [T scale and the Boltzmann constant]; news sn(20)oct [acoustic thermometers].
@ Cooling: Wu et al JLTP(11)-a1009 [laser cooling, quantum theory]; Mari & Eisert PRL(12)-a1104 + news pw(11)may [by incoherent thermal light]; Cleuren et al PRL(12) [by photons]; news pw(13)mar [solid-state refrigerator for cooling to T < 300 mK]; news NASA(14)jan [Cold Atom Lab]; Kovachy et al PRL(15) + news sn(15)apr [lensing Rb atoms to 50 pK]; news ns(17)jan [using squeezed light]; Neuhaus et al a2104 [laser cooling of a Planck-mass object].
@ Minimum temperature: Benenti & Strini PRA(15)-a1412 [in quantum thermodynamics, and dynamical Casimir effect]; Rogers PRE(17)-a1602 [and EPR paradox].
@ Maximum temperature: pbs nova(08)jan; Dai & Stojković JCAP(16)-a1601 [in a simple thermodynamical system], PRD(17)-a1704 [gas in AdS spacetime].
@ Other topics: Cercignani JSP(97) [and entropy]; Hartmann et al PRL(04)qp/03 [and subsystems]; Militello PRA(12)-a1204 [role of thermal state in dynamical regimes]; Romanelli et al PhyA-a1507 [entanglement temperature]; Ghonge & Vural JSM(18)-a1708 [as a quantum observable].