Temperature| 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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send feedback and suggestions to bombelli at olemiss.edu – modified 12
nov
2009