Biology Reference
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approximated by the center of mass and the total strength of the largest
possible cell that satisfies the closeness criterion
d
∆
<
q
,
(13)
where
d
is the diagonal of the cell currently being considered,
∆
is the
distance of particle
p
from the center of mass of that cell, and
θ
is a fixed
accuracy parameter
1. This amounts to coarse-graining clusters of
remote particles to single particles.
Based on the Barnes-Hut algorithm, Leslie Greengard and
Vladimir Rokhlin presented the fast multipole method (FMM).
110,114,115
Their formulation uses a finite series expansion of the interaction
kernel and direct cell-cell interactions in the tree. Compared to the
Barnes-Hut algorithm, this further reduces the algorithmic complexity
to
∼
O
(
N
).
7. Particle Methods for the Simulation
of Diffusion Processes
We consider the simulation of continuous spatial diffusion processes as
a simple example of biological relevance.
116
Physically, the macroscopic
phenomenon of diffusion is created by the collective behavior of a
large (in theory, infinite) number of microscopic particles, such as
molecules, undergoing Brownian motion.
116-118
From continuum
theory,
119
we can define a concentration field as the mean mass of par-
ticles per unit volume at every point in space (cf. Sec. 3.2). For abundant
diffusing particles, this allows formulating a continuous deterministic
model for the spatiotemporal evolution of the concentration field
u
(
x
,
t
) in a closed, bounded domain
Ω
. This model is formulated as
the PDE
∂
ut
t
(,)
x
⋅
=∇
((,) (, )
Dx
tu t
∇
x
for
x
∈
{
ΩΩ
∂
,
0
<
tT
≤
.
(14)
∂
∇
In this diffusion equation,
D
(
x
,
t
) denotes the diffusion tensor,
the
Nabla operator, and
∂
Ω
the boundary of the domain
Ω
.
