Interdisciplinary Applied Mathematics

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f = exp(a + b • v + cbv2),

which is known as the Maxwellian distribution and represents an equilibrium state for number density n0 and temperature T0. It can be rewritten in the familiar form

fo = n 3/2noPi exp(—e2v2),    (15.8)

where we have defined



with kB = 1.38 05 x 10 23 J/K the Boltzmann constant, and m the molecular mass.

All the macroscopic quantities are defined in terms of the distribution function: for example,


p(x,t) = m / f (x, v,t)dv

Bulk velocity



vf (x, v, t)dv


T{x,t) =    J c2 f (x, v, t)dv,

where c = v u is the peculiar velocity.

The boundary conditions take into account the wall type via the nonnegative scattering kernel, representing a probability density

R(v‘ ^ v; x; t)

that molecules hitting the wall with velocity between v and v + dv at location x at time instance t will be reflected with velocity between v and v + dv. If R is known, then we can obtain the boundary condition for the distribution function as

f (x, v,t) Vn

H (—V‘n )v‘nR(v

v; x, t)f (x, v’, t)dv’,


where H(x) is the Heaviside step function, and vn = v • n is the velocity normal to the surface. If there is no adsorption on the wall surface, then

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