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)/E; For a canonical ensemble it is then the temperature of the microcanonical ensemble composed of the system + is heat bath [@ 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; 1960's, 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 colled 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: 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; 2007, Direct experimental results have not been obtained because of the difficulty in containing a gas moving at relatvistic bulk velocities, but there is hope to get evidence from some astrophysical systems, and extensive simulations performed by 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 a0804 [inverse temperature 4-vector]; Rasinariu a0804 [moving systems appear cooler].
@ Non-equilibrium T: Essex et al AJP(03) [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.
@ Related topics: Leanhardt et al Sci(03)sep + pw(03)sep [BEC]; Beige et al qp/04-in [cooling N particles to very low T]; pbs nova(08)jan [maximum temperature?]; Liu & Wang PLA(08) [for finite number of classical spin-half particles]; > s.a. thermal radiation.

References > s.a. units [definition of K].
@ General: Ehrlich AJP(81) [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].
@ Other topics: Cercignani JSP(97) [and entropy]; Hartmann et al PRL(04)qp/03 [and subsystems].


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