Editing Negative temperature (section)

== Definition of temperature ==
The absolute [[temperature]] (Kelvin) scale can be loosely interpreted as the average kinetic energy of the system's particles. The existence of negative temperature, let alone negative temperature representing "hotter" systems than positive temperature, would seem paradoxical in this interpretation. The paradox is resolved by considering the more rigorous definition of [[thermodynamic temperature]] in terms of [[Boltzmann's entropy formula]]. This reveals the tradeoff between [[internal energy]] and [[entropy]] contained in the system, with "[[thermodynamic beta|coldness]]", the ''reciprocal'' of temperature, being the more fundamental quantity. Systems with a positive temperature will increase in entropy as one adds energy to the system, while systems with a negative temperature will decrease in entropy as one adds energy to the system.<ref>{{cite book|url=https://books.google.com/books?id=-7aP0f5O2CMC|pages= 10–14|title=The Laws of Thermodynamics: A Very Short Introduction|isbn=978-0-19-957219-9|last1=Atkins|first1=Peter W.|author-link1=Peter Atkins|date=2010-03-25|publisher= Oxford University Press|oclc=467748903}}</ref>

The definition of [[thermodynamic temperature]] {{mvar|T}} is a function of the change in the system's [[entropy]]  {{mvar|S}}  under reversible [[heat transfer]] {{math|''Q''<sub>rev</sub>}}:
: <math>T = \frac{dQ_\mathrm{rev}}{dS}.</math>
Entropy being a [[state function]], the integral of {{mvar|dS}} over any cyclical process is zero. For a system in which the entropy is purely a function of the system's energy {{mvar|E}}, the temperature can be defined as:
: <math> T  = \left( \frac{dS}{dE} \right)^{-1}.</math>

Equivalently, [[thermodynamic beta]], or "coldness", is defined as 
:<math>\beta = \frac{1}{kT} = \frac{1}{k} \frac{dS}{dE},</math>
where {{mvar|k}} is the [[Boltzmann constant]].

Note that in classical thermodynamics, {{mvar|S}} is defined in terms of temperature. This is reversed here, {{mvar|S}} is the [[statistical entropy]], a function of the possible microstates of the system, and temperature conveys information on the distribution of energy levels among the possible microstates. For systems with many degrees of freedom, the statistical and thermodynamic definitions of entropy are generally consistent with each other.

Some theorists have proposed using an alternative definition of entropy as a way to resolve perceived inconsistencies between statistical and thermodynamic entropy for small systems and systems where the number of states decreases with energy, and the temperatures derived from these entropies are different.<ref name="DunkelHilbert">{{cite journal|last1 = Dunkel|first1 = Jorn|last2 = Hilbert|first2 = Stefan|title = Consistent thermostatistics forbids negative absolute temperatures|journal = Nature Physics|volume = 10|issue = 1|pages = 67|doi = 10.1038/nphys2815|arxiv = 1304.2066|bibcode = 2014NatPh..10...67D|year = 2013|s2cid = 16757018}}</ref><ref name="HanggiHilbertDunkel">{{cite journal|last1 = Hanggi|first1 = Peter|last2 = Hilbert|first2 = Stefan|last3 = Dunkel|first3 = Jorn|title = Meaning of temperature in different thermostatistical ensembles|journal = Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences|volume = 374|issue = 2064|pages = 20150039|doi = 10.1098/rsta.2015.0039|arxiv = 1507.05713|year = 2016| pmid=26903095 | bibcode=2016RSPTA.37450039H | s2cid=39161351 }}</ref> It has been argued that the new definition would create other inconsistencies;<ref>{{Cite journal|title = Gibbs, Boltzmann, and negative temperatures|journal = American Journal of Physics|date = 2015-02-01|issn = 0002-9505|pages = 163–170|volume = 83|issue = 2|doi = 10.1119/1.4895828|first1 = Daan|last1 = Frenkel|first2 = Patrick B.|last2 = Warren|arxiv = 1403.4299 |bibcode = 2015AmJPh..83..163F |s2cid = 119179342}}</ref> its proponents have argued that this is only apparent.<ref name="HanggiHilbertDunkel"/>