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We can now apply separation of variables to obtain 2d

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Unformatted text preview: 1 a = 0.7 0.5 1.5 1 0.5 0 0 0 -0.5 -0.5 -0.5 -1 -1 -1 -2 -1.5 -1.5 -1.5 2 -1.5 -1 -0.5 0 0.5 1 1.5 -1.5 4 a = 2−1/2 -1 -0.5 0 0.5 1 1.5 a = 0.8 8 3 2 a = 0.9 0 -1 -2 -4 -2 -6 -3 -2 1.5 2 0 -1 0.5 1 4 1 0 0 6 2 1 -2 -1.5 -1 -0.5 -8 -4 -2 -1 0 1 2 -4 -3 -2 -1 0 1 2 3 4 -8 -6 -4 -2 0 2 4 6 8 Figure 5: Plots of the PDF of the decaying random walk for twelve different values of a. For each graph, 5 × 108 trials were performed. The color scheme goes from white (zero probability density), through purple and blue, to black (high probability density). M. Z. Bazant – 18.366 Random Walks and Diffusion – Problem Set 3 Solutions 9 from which we find that the velocity must satisfy 2v (c+ − c− ) = (c2 − c2 ) − + c+ + c− v= . 2 From this, we find that A=− c+ c− 2 and hence f must satisfy (c+ c− )f − f 2 + 2Df � = c+ c− . We can now apply separation of variables to obtain � 2D df 2 − (c + c ) + c c f − +− �+ 2D df (f − c+ )(f − c− ) � � � 2D df df df − − c− − c+ f − c+ c− − f 2D (− log(f − c+ ) + log(c− − f )) c− − c+ � = dz � = dz = z+k = z + k. The variable k corresponds to a translation and does not affect the form of the function; it is convenient to set k = 0. Then we find c− − f f − c+ =e c− − f = fe c− −c+ z 2D c− −c+ z 2D − c+ e c− −c+ z 2D and hence f (z ) = c− + c+ e 1+e = = 4 c− − c+ z 2D c− −c+ z 2D c− + c+ c − c+ + 2 2 � 1−e c− −c+ z 2D � c− −c+ 1 + e 2D z � � c− + c+ c− − c+ c+ − c− + tanh z. 2 2 4D Discrete versus continuous models with nonlinear drift Appendices B and C provide C++ codes to simulate the nonlinear drift problem for models A and B respectively. These codes follow an approach very similar to that stated in the problem. The largest difference is in the application of the boundary conditions. We assume that the density of particles to the left of the simulation has a constant value of ρmax /4. If we take an interval of finite length λ then the number of particles in this region should follow a Po...
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This note was uploaded on 01/23/2014 for the course MATH 18.366 taught by Professor Martinbazant during the Fall '06 term at MIT.

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