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ρmax λ/4. Similarly, to the right of the simulation, the density is assumed to have constant value of M. Z. Bazant – 18.366 Random Walks and Diﬀusion – Problem Set 3 Solutions 10 3ρmax /4, and the number of particles in an interval of length λ follows a Poisson distribution with
mean 3ρmax λ/4.
√
For model A, our steps take the form u(ρ)τ ± 2Dτ , and for the parameters in this question,
the diﬀusive term is signiﬁcantly larger than the drift. A hypothetical particle to the left of the
√
simulation region could enter the simulation region if it was in the interval −L − 2Dτ − u(ρ)τ <
x < −L, and if it took a step to the right. Since the chance of stepping right is exactly 1/2, it was
chosen to introduce particles according to a Poisson process with mean
�
√
ρmax 1 �
× × u(ρmax /4)τ + 2Dτ
4
2
√
into the interval −L < x < −L + 2Dτ + u(ρmax /4). A similar analysis shows that at the right
edge, particles need to be introduced at a rate of
�
3ρmax 1 �√
××
2Dτ − u(3ρmax /4)τ
4
2
√
into the interval L − 2Dτ + u(3ρmax /4) < x < L. For model B, assuming that the ρ is roughly
constant near the boundary, and thus ρx is roughly zero, particles take steps of size u(ρ)τ . Thus we
need to introduce no particles at the right boundary, and particles at a rate of u(ρmax /4)τ ρmax /4
at the left boundary.
When calculating the estimates of ρ and ρx at a point p in the interval, we make use of the
prescription given in the question. ρ is calculated by looking at dividing the number of particles in
the range p − l < x < p + l by 2l. ρx is calculated by ﬁnding the diﬀerence between the number of
particles in the regions p − l < x < p and p < x < p + l and dividing by l2 . If any of these counting
regions overlap with the boundary, say by an amount δ , then we make an additional contribution
to the density of ρδ .
Figures 6 and 7 show simulation output for the two models. In both cases, we see a good
agreement w...
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 Fall '06
 MartinBazant

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