Temperature| 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].
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