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What is temperature?
Peter Hänggi
Peter
Hänggi
Universität Augsburg
In collaboration with Dr. Gerhard Schmid
Temperature sensation
Temperature sensation
• Subjective temperature sensation
• Memory effects • Measuring the heat flow
h h
fl
• Heat –
H
substance ↔ Cold –
b
↔ C ld substance b
What is temperature?
Overview
I. History
II. Temperature scales
III. Thermometers
h
IV Grand Laws of Thermodynamics
IV.
G dL
f Th
d
i
V Temperature characteristics
V.
T
t
h
t i ti
Brief history of the Brief
history of the
temperature concept
temperature concept
Early temperature measurements
Early temperature measurements
No scale for measuring No
scale for measuring
temperature between some reference points Galileo Galilei
(1564 – 1642)
1592: first documented (gas‐) thermometer invented by Galileo Galilei
invented by Galileo Galilei
First thermometer & temperature scales
1638: Robert Fludd – air thermometer &scale
~1700: linseed oil thermometer by Newton
1701: red wine as temperature indicator by Rømer
1701: red wine as temperature indicator by Rømer
1702: Guillaume Amontons: Absolute zero temperature? 1714: mercury and alcohol thermometer by Fahrenheit
D fi iti
Definition of temperature scales
ft
t
l
Sir Isaac Newton
Sir
Isaac Newton Olaf Christensen
Römer
(1643 – 1727)
(1644 – 1710)
Daniel Gabriel Daniel
Gabriel René Antoine A
Anders Celsius
d C l i
Ferchault de Fahrenheit
(1701 – 1744) Réaumur
(1686 – 1736) (1683 – 1757)
Absolute Zero Ideal gas law:
p V = N kB T
V
V = 0, T = 0
◦
−273.15 C
1848: Kelvin postulates an absolute zero temperature William Thomson T
― Lord Kelvin
Lord Kelvin
(1824 – 1907) It is impossible by any procedure to reduce the temperature of It
is impossible by any procedure to reduce the temperature of
a system to zero in a finite number of operations.
Low temperature milestones
Low temperature milestones
Heike Kamerlingh Onnes
(1853 – 1926)
1908: liquid helium; 5 K
1908: liquid helium; 5 K
1995: Bose
1995:
Bose‐Einstein‐
Einstein
Condensate; 20 nK
Eric A. Cornell
Wolfgang Ketterle
Carl E. Wieman
2003: BEC; 450 pK – W. Ketterle Temperature scales
Temperature scales
Absolute temperature scale
Kelvin – scale (SI)
Reference: absolute zero temperature p
& triple point of water (273.16K; 611.73 Pa)
Scale division: 1K is equal to the fraction (273.16)‐1 of the thermodynamic temperature of the triple point of water
∆T = 1K ≡ 1˚C (degree Celsius)
Comparision of temperature scales
Kelvin [K]
Celsius [°C]
Fahrenheit [°F]
Absolute zero
0.00
‐273.15
‐459.67
Fahrenheit’s ice/salt /
mixture
255 37
255.37
‐17.78
17 78
0 00
0.00
Ice melts (at standard pressure**)
273.15
(100) = 0.00 *
32.00
Average human body temperature
309.95
36.80
98.24
Water boils (at standard pressure)
373 13
373.13
(0) = 100 (0)
= 100 *
211 97
211.97
Titanium melts
1941.00
1668.00
3034.00
source: wikipedia.org
p
g
Further temperature scales: Newton, Delisle, Réaumur, Rømer, M. Planck (linear scales)
Rudolf Plank (logarithmic scale)
* Celsius used a reversed temperature scale with “0”
0 indicating the boiling point
of water and “100” the melting point of ice
** 1 atm = 760 mmHg = 101325 Pa
Planck units
Planck units
Constant
Symbol
Value in SI units
Speed of light
c
299 792 458 m s‐1
Gravitational constant
G
6.67429 ∙ 10‐11 m3 kgg‐1 s‐2
reduced Planck’s constant
Coulomb force constant B lt
Boltzmann constant t t
Name
Planck temperature Planck length
Planck mass
Planck time
h̄
(4π²0 )−1
kB
1.054571628 ∙10‐34 J s
8 987 551 787.368 kg m3 s‐2 C‐2
23 J K
1.3806504∙10
1
3806504 10‐23
J K‐11
Expression
√
TP = h̄c5 G−1 k−2
√
lP = √h̄Gc−3
mP = √hcG
h̄cG−1
tP = h̄Gc−5
SI equivalents
1.41168∙ 1032 K
1.61625∙ 10‐35 m
2.17644∙ 10‐8 kgg
5.39124∙ 10‐44 s
Planck units
Planck units
Name
Planck temperature Planck length
Planck mass
Planck time
Planck time
Expression
√
TP = h̄c5 G−1 k−2
√
lP = √h̄Gc−3
mP = √h̄cG−1
tP = hGc
h̄G −5
SI equivalents
1.41168∙ 1032 K
1.61625∙ 10‐35 m
2.17644∙ 10‐8 kg
44 s
5.39124∙ 10
5
39124 10‐44
… ihre Bedeutung für alle Zeiten und für alle, auch
außerirdische und außermenschliche
und außermenschliche Kulturen notwendig
behalten und welche daher als natürliche Maßeinheiten
bezeichnet werden können …
… These necessarily retain their meaning for all times and for all civilizations, even extraterrestrial and non‐human ones, and can therefore be designated as natural units …
d
h f
b d i
d
l i
(Planck, 1899)
Typical temperature values [
temperature values [°C]
C]
Boiling point of Nitrogen
‐195.79
Lowest recorded surface temperature on Earth
(Vostok, Antarctica – July 21, 1983)
‐89
Highest recorded surface temperature on Earth
(Al’ Aziziyah, Libya – September 13, 1922)
58
Temperature in the Earth’s Thermosphere (80 ‐ 650 km above the surface) Melting point of diamond
~ 1500
3547
Surface temperature of the sun (photosphere)
~ 5526
Temperature in the interior of the sun
~ 15∙106
Thermometers
Gas Thermometers
Gas Thermometers
Ideall gas:
Id
µ
¶
V
T =
p
N kB
Galilei Thermometer
Volume change of water
Upon increasing T → 4°C
Water volume shrinks
Linnaeus thermometer
Carl von Linné
(1707 – 1778)
R
Reversed
d th
the C
Celsius
l i scale
l
1744: broken on delivery
1745: botanical garden in Uppsala
Anders Celsius Noise Thermometer
Noise Thermometer
O
Ohmic
Resisto
or
Johnson – Nyquist noise
V
PSDV (ω) = 2kB T R
-- classical regime only --
PSDV : power spectral density
of the voltage signal
kB : Boltzmann constant
R : resistance
Quantum fluctuation‐dissipation theorem
µ
¶
hω
h̄ω
P SDI (ω)
( ) = h̄ω coth
ReY (ω)
( )
2kB T
µ
¶
h̄ω
hω
h̄ω
hω
+
=
ReY (ω)
2
exp(βh̄ω) − 1
kB T À h̄ω
P SDI (ω) = 2kB T ReY (ω)
kB T ¿ h̄ω
P SDI (ω) = h̄ω ReY (ω)
Black body radiation
Black body radiation
8πhc
1
u(λ T ) = 5
Planck’ss law [1901]:
Planck
law [1901]: u(λ,
λ exp( λ khc T ) − 1
B
u(λ, T ) : spectral energy density
λ : wavelength
h : Planck constant
c : speed of light
kB : Boltzmann constant
Stefan – Boltzmann law:
E ∝ T4
Thermometer !
Cosmic background temperature
Cosmic background temperature
T = 2.725 ± …. K Four Grand Laws of Four
Grand Laws of
Thermodynamics
Zeroth Law
Zeroth Law
Transitivity !
A in equilibrium with B:
fAB (pA , VA ; pB , VB , ...) = 0
B in equilibrium with C:
fBC (pB , VB ; pC , VC , ...) = 0
⇒ A in equilibrium with C ⇔ fAC (pA , VA ; pC , VC , ...)) = 0
Allows the formal introduction of a temperature:
T = TA (pA , VA ; ...) = TB (pB , VB ; ...) = TC (pC , VC ; ...)
First Law
First Law
Julius Robert von Mayer
von Mayer
(1814 – 1878)
James Prescott Joule
(1818 – 1889)
Hermann von Helmholtz
(1821 – 1894)
First Law – Energy Conservation
First Law Energy Conservation
∆E = ∆Q + ∆W
∆E
change in internal energy
∆Q heat
h t added
dd d on the
th system
t
∆W
work done on the system
H. von Helmholtz: “Über die Erhaltung der Kraft” (1847)
∆E = (T ∆S)quasi-static − (p∆V )quasi-static
Second Law
Second Law
Rudolf Julius Emanuel Clausius
d lf li
l l i
(1822 – 1888)
William Thomson alias Lord Kelvin
(1824 – 1907)
Heat generally cannot
Heat generally cannot No cyclic process exists whose sole
No cyclic process exists whose sole spontaneously flow from a effect is to extract heat from a material at lower temperature to single heat bath at temperature T a material at higher temperature.
i l hi h
and convert it entirely to work. d
i
i l
k
δQ = T dS
(Zürich, 1865)
Perpetuum mobile of the second kind ?
One heat reservoir
NO !
Two heat reservoirs
Entropy S – content of transformation
„Verwandlungswert“
V
dl
t“
d S = δQ
rev
V2 , T 2
Γrev
T;
I
Γirrev
δQ
irrev
rev
< δQ
δQ
≤0
T
C
C = Γ−1
rev + Γirrev
V1 , T1
Z
δQ
T
Γ
Z irrev
δQ
S(V2 , T2 ) − S(V1 , T1 ) =
Γrev T
S(V2 , T2 ) − S(V1 , T1 ) ≥
∂S
≥ 0 NO !
∂t
Thermodynamic potentials
Thermodynamic potentials
Internal energy: E(S, V, {Ni }) = T S − p V +
Helmholtz free energy: F (T, V, {Ni }) = E − T S
X
μi Ni
Enthalpy: H(S, p, {Ni }) = E + p V
Gibbs free energy: G(T, p, {Ni }) = E + p V − T S =
Planck potential: Φ(T, p, {Ni }) = −G/T
X
μi Ni
Third Law
Third Law
… and Planck’ss version
… and Planck
version
Approaching zero temperature
Approaching zero temperature
Famous exception of the 3
Famous
exception of the 3rd law
Temperature definitions
Temperature definitions
Thermodynamic Temperature
e ody a
e pe a u e
δQ
rev
= T dS ← thermodynamic entropy
S = S(E, V, N1 , N2 , ...; M, P, ...)
S(E ...):
S(E,
) (continuous)
( ti
) & differentiable
diff
ti bl and
d
monotonic function of the internal energy E
µ
∂S
∂E
¶
...
1
=
T
How to determine the absolute thermodynamic temperature
h
d
T
log
=
T0
Z
t
t0
¡ dv ¢
dt p
v + c0p
³
dt
dt
dp
t : arbitrary temperature scale
´
v : volume
c0p : specific heat at constant pressure
“ ... wo nun wieder unter dem Integralzeichen lauter direkt und
g
verhältnismäßig bequem meßbare Größen stehen. ...” M. Planck
A statistical definition of Temperature
A statistical definition of Temperature
Ω(E,
( , V,, N ) : number of microstates for a ggiven macrostate ((E,, V,, N )
E1 , V1 , N1 E2 , V2 , N2
E1 , V1 , N1 E2 , V2 , N2
Ω1 (E1 , V1 , N1 ) Ω2 (E2 , V2 , N2 ) ⇒ Ω = Ω1 · Ω2
∂Ω1
∂Ω2
in equilibrium: δΩ =
δE1 Ω2 + Ω1
δE2 = 0
∂E1
∂E2
1 ∂Ω1
1 ∂Ω2
energy conservation:
ti
δE1 = −δE
δE2 ⇒
=
Ω1 ∂E1
Ω2 ∂E2
Boltzmann (Planck): S = kB lnΩ
1
1
⇒
=
⇒ T1 = T2 = T
T1
T2
Temperature Fluctuations –
An Oxymoron
Ch. Kittel, Physics Today, May 1988, p. 93
Ch. Kittel, Physics Today, May 1988, p. 93
Nano‐system
BATH
1
β=
kB T
T does NOT fluctuate!
|∆U | |∆β| = 1 ↔ |∆U | |∆T | = kB T 2
“…Temperature
p
NO!
is precisely
p
y defined onlyy for a system
y
in thermal
equilibrium with a heat bath: The temperature of a system A, however
small, is defined as equal to the temperature of a very large heat
reservoir B with which the system is in equilibrium and in thermal
contact. Thermal contact means that A and B can exchange energy,
although insulated from the outer World. …” Molecular Dynamics y
• Every quadratic term in the Hamiltonian 1
contributes 2 kB T to the internal energy
contributes to the internal energy
• Classical systems: VIRIAL THEOREM
[Hi ∝ qin or pni ⇒ hEi i = kB T /n]
• Classical harmonic Oscillator
hp2i i
1
= mhvi2 i =
2m
2
1
mω 2 hx2i i =
2
1
kB T
2
1
kB T
2
NOT SO FOR A QUANTUM OSCILLATOR
Specific Heat
E=
X
Ei exp(−βEi )
i
±X
exp(−βEi )
i
dE
C =
= (kB T 2 )−1 h(Ei − hEi i)2 i> 0 !
dT
E
∂S
= 1/T ,
Note also:
∂E
S
∂2S
1
=- E 2
2
∂E
C T
CE<0
CE>0
E
Specific heat paradox (microcanonical)
R. E
R
Emden
d (1907)
(1907); A
A.S.
S Eddi
Eddington
t (1926)
(1926);
D. Lynden-Bell & R. Wood (1968); W. Thirring (1970);
W. Thirring, H. Narnhofer & H. A. Posch (2003)
Coulomb: hΦi i = −kB T
CLASSICAL !
1
Tkin = − hΦtot i = 3N kB T/2
2
2Tkin + hΦtot i = 0 = E + Tkin
C
E
dE
dTkin
=−
= −3N kB /2 < 0
=
dT
dT
Quantum Regime
Quantum Regime
h = 6.62606896 · 10
−34
4
Js= 2
KJ R K
with KJ = 2e/h
/ , RK = h/e
/
2
hν = 1 kg c2
→ ν = 135639273 · 1042 Hz
Quantum Harmonic Oscillator
Quantum Harmonic Oscillator
p2
M 2 2
HS =
+
ω0 x ,
2M
2
1
En = (n + )h̄ω0 ,
2
1
β=
kB T
∂E
Specific heat: C =
Specific heat: ,
∂T
E
h̄ω0
h̄ω0
E=
+
2
exp (h̄ω0 β) − 1
Quantum harmonic Oscillator:
Quantum harmonic Oscillator:
the limit ω0 → 0 does NOT yield the specific heat C E = kB /2
of the free particle
1
kB T
E=
6
2
No virial
N
i i l th
theorem !
No equipartition th.!
? Negative Temperature ?
? Negative Temperature ?
~ | = 1/2
Spin system: p y
|S|
/ ;
~;
μ
~ = γS
H=−
X
~
μ
~i · B
1
1
~
~
S k B ⇒ Two-State-System:
Two State System: ²g = − γB < ²e = + γB = μB
2
2
N = ng + ne & E = μ
μB(n
( e − ng ) , typically
yp
y E<0
µ
¶
E
1
N−
ng =
2
μB
µ
¶
1
E
ne =
N+
2
μB
N!
Ω=
⇒ S = kB ln Ω
ng ! ne !
1
∂S
⇒
=
T
∂E
Negative (Spin)‐Temperature!
Relativistic Regime
Relativistic Regime
0
Σ(u = 0) −→
→ Σ (u 6= 0)
x0 = γ(u) (x − u · t)
¡
¢
0
2
t = γ(u) t − ux/c
µ
¶ /
2 −1/2
u
with γ(u) = 1 − 2
c
x2 − c2 t2 = (x
( 0 )2 − c2 (t
( 0 )2
Note: thermodynamic observables are NONLOCAL
Note: thermodynamic observables are NONLOCAL
J. Dunkel & P.H., Relativistic Brownian motion, Phys. Rep. 471, 1 - 73 (2009)
Maxwell‐Boltzmann
Maxwell
Boltzmann distribution
distribution
velocity
y distribution:
f~v (vx , vy , vz ) =
µ
m
2π kB T
¶3/2
Ã
vx2
vy2
vz2
+ +
exp −
2 kB T /m
25 °C
James Clerk Maxwell
(1831 – 1879) ! v < c violated !
Ludwig Eduard Boltzmann
(1844 – 1906)
!
Relativistic Brownian motion
J. Dunkel, P.H., Phys. Rep. 471, 1 (2009)
l0 (u) = l0 γ −1
t0 (u) = t0 γ +1
T 0 (u) = T0 γ −α
Masoliver & Weiss, Eur. J. Phys. 17: 190 (1996)
Relativistic thermodynamics
Relativistic thermodynamics
Jüttner Gas
d/2
¡
¢
2
exp −βp /2m
fMaxwell (~
p) = [β/(2πm)]
h
i
fJüttner (p)
(~) = Zd−11 exp −β
βJ (m
( 2 c4 + p2 c2 )1/2
u=0
h~
p · ~v i = dkB T = d/βJ
u=0
statistical relativistic temperature
T = T = (k
( B βJ )−11
J. Dunkel & P.H., Phys. Rep. 471, 1-73 (2009)
Moving Observers
Σ(u = 0) −→ Σ0 (u 6= 0)
0 3
£
¤
mγ(v
)
−1
0
0
0
2
0
exp −βJ γ(u)mγ(v )(c + uv )
fJüttner (v , u) = Z1
γ(u)
v0 : velocity in moving frame
u = 0.2c
u = 0.2c
J. Dunkel & P.H., Phys. Rep. 471, 1-73 (2009)
Measuring temperature in Lorentz invariant way
Lorent invariant
Lorentz
in ariant equipartition theorem
3
0
0
2
kB T = mγ(u)
γ( ) hγ(
hγ(v )(
)(v + u)) it0
with u = −hv 0 it0
u/c
u/c
Take Home Messages
• ∃ absolute temperature T ≥ 0
• T =
S
∂E
∂S ...
with S(E) a concave function of E !
E
• S(E) : microcanonical NOT always = S(E) : canonical
( long range,
range non-ergodic
non ergodic parts )
• T does NOT fluctuate
• quantum mechanics: no virial theorem; no equipartion
• relativistic statistical thermometer
TΣ = TΣ0