problemset3

# For model b assuming that the is roughly constant

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Unformatted text preview: isson distribution with mean 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 &lt; x &lt; −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 &lt; x &lt; −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) &lt; x &lt; 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 &lt; x &lt; p + l by 2l. ρx is calculated by ﬁnding the diﬀerence between the number of particles in the regions p − l &lt; x &lt; p and p &lt; x &lt; 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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