# Boltzman equation

July 11, 2012, 4:50 am
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Non-equilibrium heat flow from (unconventially) colored hot to cold. Source: CC-sa-unported

The Boltzmann equation, in physics, describes the statistical behavior of a fluid wich is not in thermodynamic equilibrium. That is to say, the Boltzman equation applies where there are spatial temperature gradients, making heat flow away from warmer areas of the fluid (or gas) to cooler ones. This heat flow occurs by the random transport of fluid or gaseous particles. This formalism was first developed by Ludwig Boltzmann in the year 1872

The Navier-Stokes equations can be derived from the Boltzmann equation, also termed the Boltzmann transport equation, by considering appropriate limits, i.e. Knudsen and Mach numbers tending to zero, and appropriate averaging procedures to define new coarse-grained variables (velocity and pressure) and associated transport coefficients (viscosity and density).

 This article is written at a definitional level only. Authors wishing to expand this entry are inivited to expand the present treatment, which additions will be peer reviewed prior to publication of any expansion.

• Physical Oceanography Index
• M. Farge, N. Kevlahan, V. Perrier, and E. Goirand. Wavelets and turbulence. Proc. IEEE, 84:639–669, 1996.
• Leif Arkeryd. 1972. On the Boltzmann equation part I: Existence. Arch. Rational Mech. Anal. 45: 1–16.
• R.J.DiPerna and P.-L.Lions. 1989. On the Cauchy problem for Boltzmann equations: global existence and weak stability. Ann. of Math. (2) 130: 321–366.
Glossary