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 B 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 (or Celcius) 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

Gas Pressure Thermometer Ice point LN2 Steam point

Gas Pressure Thermometer Celsius scale P = a[T(oC) + 273.15] Ice point Steam point LN2

Phase diagram of water The single combination of pressure and temperature at which liquid water, solid ice, and water vapour can coexist in a stable equilibrium occurs at exactly 273.16 K (0.01 °C) and a partial vapour pressure of 611.73 pascals (ca. 6.1173 millibars, 0.0060373057 atm). At that point, it is possible to change all of the substance to ice, water, or vapor on making arbitrarily small changes in pressure and temperature. Note that even if the total pressure of a system is well above triple point of water, provided the partial pressure of the water vapour is 611.73 pascals then the system can still be brought to the triple point of water. Strictly speaking, the surfaces separating the different phases should also be perfectly flat, to abnegate the effects of surface tensions. Water has an unusual and complex phase diagram, although this does not affect general comments about the triple point. At high temperatures, increasing pressure results first in liquid and then solid water. (Above around 109 Pa a crystalline form of ice forms that is denser than liquid water.) At lower temperatures under compression, the liquid state ceases to appear, and water passes directly from gas to solid. At constant pressures above the triple point, heating ice causes it to pass from solid to liquid to gas, or steam, also known as water vapour. At pressures below the triple point, such as those that occur in outer space, where the pressure is near zero, liquid water cannot exist. In a process known as sublimation, ice skips the liquid stage and becomes steam when heated. Near triple point can have ice, water, or vapor on making arbitrarily small changes in pressure and temperature.

Concept of Absolute Zero (1703) Guillaume Amonton first derived mathematically the idea of absolute zero based on Boyle-Mariotte’s law in 1703. For a fixed amount of gas in a fixed volume, p = kT Amonton’s absolute zero ≈ 33 K

Other Types of Thermometer Metal resistor : R = aT + b Semiconductor : R = R0exp(-Eg/kBT) Thermocouple : E = aT + bT2 Low Temperature Thermometry

Platinum resistance thermometer

CERNOX thermometer Insensitive to magnetic fields !

International Temperature Scale of 1990 1. Units of Temperature The unit of the fundamental physical quantity known as thermodynamic temperature, symbol T, is the kelvin, symbol K, defined as the fraction 1/273.16 of the thermodynamic temperature of the triple point of water1. Because of the way earlier temperature scales were defined, it remains common practice to express a temperature in terms of its difference from 273.15 K, the ice point. A thermodynamic temperature, T, expressed in this way is known as a Celsius temperature, symbol t, defined by: t / °C = T/K - 273.15 .     (1) The unit of Celsius temperature is the degree Celsius, symbol °C, which is by definition equal in magnitude to the kelvin. A difference of temperature may be expressed in kelvins or degrees Celsius. The International Temperature Scale of 1990 (ITS-90) defines both International Kelvin Temperatures, symbol T90, and International Celsius Temperatures, symbol t90. The relation between T90 and t90, is the same as that between T and t, i.e.: t90 / °C = T90/K - 273.15 .     (2) The unit of the physical quantity T90 is the kelvin, symbol K, and the unit of the physical quantity t90, is the degree Celsius, symbol °C, as is the case for the thermodynamic temperature T and the Celsius temperature t.

Bernoulli's choice: Heads or Tails? N = # of slots, # of macrostates  = multiplicity, # of microstates Pascal’s triangle N 1 2 3 4 5 =2N 20 21 22 23 24 25 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 Example: For N=4 fair coin tosses there are N+1=5 macrostates each containing n heads where n = 0, 1, 2, 3, 4. Each macrostate has 4Cn occurrences of n heads with a total # of microstates equal to the multiplicity .

16 different configurations (microstates), 5 different macrostates Prob. (microstate) Macrostates: n,m Macrostate: n-m hhhh 1/16 4, 0 4 thhh 3, 1 2 hthh hhth hhht tthh 2, 2 thth htht hhtt htth thht httt 1, 3 -2 thtt ttht ttth tttt 0, 4 -4 16 different configurations (microstates), 5 different macrostates

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

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)

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’

Microcanonical ensemble: An ensemble of snapshots of a system with the same N, V, and E A collection of systems that each have the same fixed energy. E (E)

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 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

Canonical ensemble: P()  (E-)1  exp[-/kBT] Log10 (P()) 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’