Lecture Notes in Computational Science and Engineering - Bioserver

338 Steffen Seeger et al.
�
(Φ) s (t,x)= dc Φ(t,x,c)fs(t,x,c) . (4)
Multiplying (2) with Φ and integrating over c we obtain [8] a balance equation
for the quantity (Φ) s (also known as Enskog’s general equation of change)
∂ t(Φ) s + ∂ x · (c Φ) s =
�
�
∂tΦ + ∂x · cΦ + ∂c · asΦ +
s
�
�
r∈Ns
dc ΦSrs , (5)
where we made use of partial integration and the property fs → 0for�c� →∞
due to (1).
2.3 The Maxwell gas model
Throughout this work we will assume the particular interaction model of socalled
Maxwell molecules, for which the particles repel each other with a
force inversely proportional to the (2d − 1)th power of their distance. As
has been known for quite a long time [9], this simplifies the collision integral
considerably. Since the Boltzmann-type collision operator (3) is similar to
a convolution integral, the simplifications are even greater when employing
a Fourier-transformation with regard to particle velocity. This method of
transformation of the Boltzmann equation into an equation for the characteristic
function [10] has been proposed and extensively studied by Bobylev
et al. [11] with a review of the method and results given in [12]. It can be
applied also for other interaction models (i.e. hard spheres and the BGK approximation)
but here we restrict our considerations to Maxwell interaction.
Setting Φ = 1
(2 π) d/2 e iχ·c we find the associated (Φ) s to be the characteristic
function ϕs(t,x,χ) with the equation of motion
∂tϕs + ∂x · ∂iχ ϕs = Γs + �
Ξrs[ϕr,ϕs],
collision term Ξrs[ϕr,ϕs] = 1
(2π) d/2
and force term Γs[ϕs] = 1
(2 π) d/2
r∈Ns
� dc Srs e iχ·c
� dc fs ∂ c · a s e iχ·c .
After a straightforward calculation following the idea of Bobylev [11] we find
Ξ2D �
2 κrs
rs [ϕr,ϕs] = µrs (2π)2
�
dε Ω[ϕr,ϕs]
Ξ 3D
rs [ϕr,ϕs] =
� 2 κrs
µrs (2π)3
�
dεdϕ εΩ[ϕr,ϕs]
for the 2D and 3D Maxwell gas respectively. Here ε is a dimensionless collision
parameter, ϕ (without any index) is the tilt of the scattering plane and
(6)
(7)