View on Cold in 17th Century

…while the sources of heat were obvious – the sun, the crackle of a fire, the life force of animals and human beings

– cold was a mystery without an obvious source, a chill associated with death, inexplicable, too fearsome to

investigate.

“Absolute Zero and the Conquest of Cold” by T. Shachtman

• Heat “energy in transit” flows from hot to cold: (Thot > Tcold)

• Thermal equilibrium “thermalization” is when Thot = Tcold

•Arrow of time, irreversibility, time reversal symmetry breaking

Zeroth law of thermodynamics

A C

B C

Diathermal wall

If two systems are separately in thermal equilibrium with a third system, they are in thermal equilibrium with each other.

C can be considered the thermometer. If C is at a certain temperature then A and B are also at the same temperature.

Simplified constant-volume gas thermometer

Pressure (P = gh) is the thermometric property that changes with temperature and is easily measured.

Temperature scales

• Assign arbitrary numbers to two convenient temperatures such as melting and boiling points of water. 0 and 100 for the celsius scale.

• Take a certain property of a material and say that it varies linearly with temperature.

X = aT + b

• For a gas thermometer:

P = aT + b

-300 -200 -100 0 100 200

-273.15 oCPr

essu

re

Temperature (oC)

Gas Pressure ThermometerGas Pressure Thermometer

Steam point

Ice point

LN2

P = aP = a[[TT((ooC)C) + + 273.15]273.15]

Gas Pressure ThermometerGas Pressure Thermometer

Celsius scale

Steam point

Ice point

LN2

-300 -200 -100 0 100 200

-273.15 oCPr

essu

re

Temperature (oC)

Phase diagram of water

Near triple point can have ice, water, or vapor on making arbitrarily small changes in pressure and temperature.

Guillaume Amonton first derived mathematically the idea of absolute zero based on Boyle-Mariotte’s law in 1703.

Concept of Absolute Zero(1703)

Amonton’s absolute zero ≈ 33 K

For a fixed amount of gas in a fixed volume,

p = kT

Other Types of ThermometerOther Types of Thermometer

•Metal resistor : R = aT + b•Semiconductor : logR = a blogT•Thermocouple : = aT + bT2

Low Temperature ThermometryLow Temperature Thermometry

0 50 100 150 200 250 300 350 4000

50

100

150

R (

)

T (K)

Platinum resistance thermometer

0 100 200 300 400

100

1000

10000

R (

)

T (K)

CERNOX thermometer

International Temperature Scale of 1990

16 different configurations (microstates), 5 different macrostates

microstate Prob. (microstate) Macrostates: n,m Macrostate: n-m

hhhh 1/16 4, 0 4

thhh 1/16 3, 1 2

hthh 1/16 3, 1 2

hhth 1/16 3, 1 2

hhht 1/16 3, 1 2

tthh 1/16 2, 2 0

thth 1/16 2, 2 0

htht 1/16 2, 2 0

hhtt 1/16 2, 2 0

htth 1/16 2, 2 0

thht 1/16 2, 2 0

httt 1/16 1, 3 -2

thtt 1/16 1, 3 -2

ttht 1/16 1, 3 -2

ttth 1/16 1, 3 -2

tttt 1/16 0, 4 -4

Microcanonical ensemble:

• Total system ‘1+2’ contains 20 energy quanta and 100 levels.• Subsystem ‘1’ containing 60 levels with total energy x is in equilibrium with subsystem ‘2’ containing 40 levels with total energy 20-x.• At equilibrium (max), x=12 energy quanta in ‘1’ and 8 energy quanta in ‘2’

Ensemble: All the parts of a thing taken together, so that each part is considered only in relation to the whole.

The most likely macrostate the system will find itself in is the one with the maximum number of microstates.

E1

1(E1)

E2

2(E2)

TkdEd

dEd

B

1lnln

2

2

1

1

Most likely macrostate the system will find itself in is the one with the maximum number of microstates. (50h for 100 tosses)

0

2e+028

4e+028

6e+028

8e+028

1e+029

1.2e+029

0 20 40 60 80 100xMacrostate

Num

ber o

f Mic

rost

ates

()

E

(E)

Microcanonical ensemble: An ensemble of snapshots of a system with the same N, V, and E

A collection of systems thateach have the same fixed energy.

Canonical ensemble: An ensemble of snapshots of a system with the same N, V, and T (red box with energy << E. Exchange of energy with reservoir.

E-

(E-)

I()

1 1

1

1

1

1

1

11

1

1 1

1

1

1

1

1 1

1

1

1 1

1

1

1

1

1

1

1

1

1

1

1 1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1 1

1

1

1

1

1

1

1 1

1

1

1

11

1

1

1

1

1

1

1

1

1

1 1

1

1

1

1

1

1

1

1

1

1 1

1

11

1

1

1

1 1

1

1

1

1

1

1

1

11

1

Canonical ensemble: P() (E-)1 exp[-/kBT]

• Total system ‘1+2’ contains 20 energy quanta and 100 levels.• x-axis is # of energy quanta in subsystem ‘1’ in equilibrium with ‘2’• y-axis is log10 of corresponding multiplicity of reservoir ‘2’

Log 1

0 (P

())