Tensors for Physics

12.3 Averages Over Velocity Distributions 221

Thethirteen moments approximationof kinetic gas theory [45] employs just these 3

plus 5 moments describing the heat flux and the friction pressure tensor, in addition

to the 5 variables associated with the conserved quantities, viz. the number density

n, the temperatureTand average flow velocityv.

In addition to the heat conductivity and the viscosity, also a contribution to the

symmetric traceless part of the pressure tensor which is proportional to the gradient of

theheat fluxqandthus proportional tothesecondspatial derivativeof thetemperature

Tis described by the thirteen moments approximation. This phenomenon, where one

has

pμν∼−∇μqν ∼∇μ∇νT,

was already predicted by Maxwell, it is referred to asMaxwell’s thermal pressure.

An experimental manifestation of this effect is provided by light-induced velocity

selective heating or cooling in gases, cf. [46]. For some applications, however, more

than thirteen moments have to be included for the solution of the Boltzmann equation

[41, 47].

A side remark: in dilute gases and in the hydrodynamic regime, the heat flux

and the viscous friction pressure tensor are independent of the densityn. The factor

(nkBT)−^1 in (12.96) implies that the deviationΦfrom the Maxwell velocity distri-

bution is proportional ton−^1 , thus it is the larger the smaller the densitynis. At a

lower density, fewer particles have to ‘work’ harder to transport the energy and the

linear momentum. Maxwell’s thermal pressure is also proportional ton−^1.

12.3 Exercise: Second Order Contributions of the Kinetic Heat Flux and
Friction Pressure Tensor to the Entropy
The ‘non-equilibrium’ entropy, per particle, associated with the velocity distribution
functionf=fM( 1 +Φ), is given bys=−kB〈ln(f/fM)〉=−kB〈( 1 +Φ)ln( 1 +Φ)〉M,
wherefMis the local Maxwell distribution andΦis the deviation off fromfM.
By analogy with (12.39), the contribution up to second order in the deviation is
s=−kB^12 〈Φ^2 〉M.
Determine the second order contributions to the entropy associated with heat flux
and the symmetric traceless pressure tensor.

Structure Factor 12.4 Anisotropic Pair Correlation Function and Static

12.4.3 Static Structure Factor

The particles surrounding any given reference particle in a dense fluid, in a liquids

or a colloidal solution, possess a short ranged order which is also referred to as the

structure of a liquid. This property is characterized by thepair correlation function

or by thestatic structure factor. For fluids composed of spherical particles, these

functions are isotropic, in thermal equilibrium. In non-equilibrium situations, these