Editing Kinetic theory of gases (section)

== Detailed balance ==

=== Fluctuation and dissipation ===
{{Main|Fluctuation-dissipation theorem}}
The kinetic theory of gases entails that due to the [[microscopic reversibility]] of the gas particles' detailed dynamics, the system must obey the principle of [[detailed balance]]. Specifically, the [[fluctuation-dissipation theorem]] applies to the [[Brownian motion]] (or [[diffusion]]) and the [[Drag (physics)|drag force]], which leads to the [[Einstein relation (kinetic theory)|Einstein–Smoluchowski equation]]:<ref>{{ cite book | last1 = Dill | first1 = Ken A. | url = https://books.google.com/books?id=hdeODhjp1bUC&pg=PA327 | title = Molecular Driving Forces: Statistical Thermodynamics in Chemistry and Biology | last2 = Bromberg | first2 = Sarina | date = 2003 | publisher = Garland Science | isbn = 9780815320517 | pages = 327 | language = en}}</ref>
<math display="block"> D = \mu \, k_\text{B} T, </math> where
* {{mvar|D}} is the [[mass diffusivity]];
* {{mvar|μ}} is the "mobility", or the ratio of the particle's [[Terminal velocity|terminal]] [[drift velocity]] to an applied [[force]], {{math|1=''μ'' = ''v''<sub>d</sub>/''F''}};
* {{math|''k''<sub>B</sub>}} is the [[Boltzmann constant]];
* {{mvar|T}} is the [[absolute temperature]].
Note that the mobility {{math|1=''μ'' = ''v''<sub>d</sub>/''F''}} can be calculated based on the viscosity of the gas; Therefore, the Einstein–Smoluchowski equation also provides a relation between the mass diffusivity and the viscosity of the gas.

=== Onsager reciprocal relations ===
{{Main|Onsager reciprocal relations}}
The mathematical similarities between the expressions for shear viscocity, thermal conductivity and diffusion coefficient of the ideal (dilute) gas is not a coincidence; It is a direct result of the [[Onsager reciprocal relations]] (i.e. the detailed balance of the [[Microscopic reversibility|reversible dynamics]] of the particles), when applied to the [[convection]] (matter flow due to temperature gradient, and heat flow due to pressure gradient) and [[Advection#Distinction between advection and convection|advection]] (matter flow due to the velocity of particles, and momentum transfer due to pressure gradient) of the ideal (dilute) gas.