# Temperature

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The temperature of a system in classical thermodynamics is intimately related to the zeroth law of thermodynamics; two systems having to have the same temperature if they are to be in thermal equilibrium (i.e. there is no net heat flow between them). However, it is most useful to have a temperature scale. By making use of the ideal gas law one can define an absolute temperature

$T = \frac{pV}{Nk_B}$

however, perhaps a better definition of temperature is

$\frac{1}{T(E,V,N)} = \left. \frac{\partial S}{\partial E}\right\vert_{V,N}$

where $S$ is the entropy. That said, in the words of Landau and Lifshitz "Like the entropy, the temperature is seen to be a purely statistical quantity, which has meaning only for macroscopic bodies" [1]. For small systems, where fluctuations become significant, things become more complicated [2] [3].

## Temperature scale

Temperature has the SI units (Système International d'Unités) of kelvin (K) (named in honour of William Thomson, Baron Kelvin of Largs [4]) The kelvin is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water[5] [6].

#### Non-SI temperature scales

Rankine temperature scale
0°R corresponds to 0 kelvin, and 1.8 degrees Rankine is equivalent to 1 kevlin [7]. The Rankine temperature scale is named after William John Macquorn Rankine.

## Kinetic temperature

$T = \frac{2}{3} \frac{1}{k_B} \overline {\left(\frac{1}{2}m_i v_i^2\right)}$

where $k_B$ is the Boltzmann constant. The kinematic temperature so defined is related to the equipartition theorem; for more details, see Configuration integral.

## Inverse temperature

It is frequently convenient to define a so-called inverse temperature, $\beta$, such that

$\beta := \frac{1}{k_BT}$

## References

1. L. D. Landau and E. M. Lifshitz "Statistical Physics", Course of Theoretical Physics volume 5 Part 1 3rd Edition (1984) ISBN 0750633727 p. 35
2. Richard McFee "On Fluctuations of Temperature in Small Systems", American Journal of Physics 41 pp. 230-234 (1973)
3. Grey Sh. Boltachev and Jürn W. P. Schmelzer "On the definition of temperature and its fluctuations in small systems", Journal of Chemical Physics 133 134509 (2010)
4. William Thomson "On an Absolute Thermometric Scale, founded on Carnot's Theory of the Motive Power of Heat, and calculated from the Results of Regnault's Experiments on the Pressure and Latent Heat of Steam", Philosophical Magazine October pp. (1848)
5. H. Preston-Thomas "The International Temperature Scale of 1990 (ITS-90)", Metrologia 27 pp. 3-10 (1990)
6. H. Preston-Thomas "ERRATUM: The International Temperature Scale of 1990 (ITS-90)", Metrologia 27 p. 107 (1990)
7. NIST guide to SI Units
8. Hans Henrik Rugh "Dynamical Approach to Temperature", Physical Review Letters 78 pp. 772-774 (1997)
9. András Baranyai "On the configurational temperature of simple fluids", Journal of Chemical Physics 112 pp. 3964-3966 (2000)
10. Alexander V. Popov and Rigoberto Hernandez "Ontology of temperature in nonequilibrium systems", Journal of Chemical Physics 126 244506 (2007)
11. J.-L. Garden, J. Richard, and H. Guillou "Temperature of systems out of thermodynamic equilibrium", Journal of Chemical Physics 129 044508 (2008)
12. Norman F. Ramsey "Thermodynamics and Statistical Mechanics at Negative Absolute Temperatures", Physical Review 103 pp. 20-28 (1956)