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Greens function and is useful for finding

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Green’s function, and is useful for finding concentration profiles for more complicated initial conditions. In particular, consider that the initial con- centration distribution is given by c ( x, 0) = δ ( x ), where δ ( x ) is the Dirac delta function and basically means that there is a spike at the origin. In particular, you will show that G ( x, t ) = 1 4 πDt e - x 2 4 Dt , (2) where we introduce the notation G ( x, t ) to signify that this is the concentra- tion profile for the special case in which the initial concentration is the spike at the origin as represented by the delta function. To obtain the solution, we will Fourier transform the diffusion equation in the spatial variable x according to the Fourier transform convention ˜ f ( k ) = 1 2 π Z -∞ f ( x ) e - ikx dx, (3) and f ( x ) = Z -∞ ˜ f ( k ) e ikx dk. (4) Using these definitions, Fourier transform both sides of the diffusion equation to arrive at the ordinary differential equation d ˜ c ( k, t ) dt = - Dk 2 ˜ c ( k, t ) . (5) Solve this differential equation to obtain ˜ c ( k, t ) and make sure to use the initial condition c ( x, 0) = δ ( x ) to find ˜ c ( k, 0). Then invert the Fourier trans- form on ˜ c ( k, t ) to find c ( x, t ). NOTE: You will need to use completion of 2
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the square to carry out the inversion. Make sure you explain all of your steps. We are big on having you not only do the analysis correctly, but also to explain what you are doing and why you are doing it. Also, explain why I said this is the solution for “free space”. Why would this solution fail to describe diffusion in a finite box? (b) Using the solution we obtained above, find h x i and h x 2 i . In general, we have that h x n i = R -∞ x n c ( x, t ) dx R -∞ c ( x, t ) dx . (6) Explain what you find for both the first and second moments of the distribu- tion as a function of time and explain how it relates to the estimated diffusion time t = L 2 /D which we use to find the time scale for diffusion over a length L . Using the Einstein-Stokes relation given by D = k B T 6 πηa , (7) where η is the viscosity which for water is η water = 10 - 3 Pa s and a is the radius of the diffusing particle, estimate the diffusion constant for a protein in water and make a log-log plot of diffusion time vs distance (with distances ranging from 1 nm to 1 m) and comment on its biological significance. Also, make a plot of the solution for the point source as a function of time by showing c ( x, t ) at various times t using the same diffusion constant.
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