Temperature  

In General > s.a. thermodynamics.
* 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.
$ 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 + is 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].
* Values: 0 K = –273ºC; 1960s, The lowest T's 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.

Specific Systems and Effects > s.a. black-hole thermodynamics; ising model [roughening T]; quantum fields in curved spacetime.
* Negative T: 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, so 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 nonrelativistic 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.
@ Microcanonical T: Davis & Blakie JPA(05)cm [classical Bose gas].
@ Negative T: Lavenda JPA(99) [argument against].
@ 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 [not well defined]; Mi et al MPLA(09) [and black-body radiation]; Mitchell & Petrov EJP(09) [moving medium].
@ Non-equilibrium T: Essex et al AJP(03)oct [radiation]; Bertin et al PRL(04) [lattice, with conserved energy]; Carati PhyA(06).
@ 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].
@ Cold matter: Leanhardt et al Sci(03)sep + pw(03)sep [BEC at 500 pK]; Beige et al BJP(05)qp/04-in [cooling N particles to very low T]; news bbc(09)jul [Planck observatory at 0.1 K]; Stamper-Kurn Phy(09); > see condensed matter [supercooled liquids]; molecules [ultracold].
@ Related topics: pbs nova(08)jan [maximum temperature?]; > s.a. thermal radiation.

References > s.a. units [definition of K].
@ General: Ehrlich AJP(81)jul [concept]; Beghian NCB(93); Rugh PRL(97) [dynamical approach]; Shachtman 99 [history; I].
@ Limitations of concept: news Nat(04)aug [meaningless for nanotubes]; Hartmann & Mahler EPL(05)cm/04 [spin-1 chain]; Hartmann CP(06)cm [local T, minimum length scales]; García-Sáez et al PRA-a0808 [local T, quantum].
@ Other topics: Cercignani JSP(97) [and entropy]; Hartmann et al PRL(04)qp/03 [and subsystems].


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