I thought I would do a little physics for fun, since it has been over 20 years since I last did any. I picked up Sethna's Stat Mech book. He gives a derivation of the diffusion equation that goes as follows:(adsbygoogle = window.adsbygoogle || []).push({});

A particle makes random steps, ie x(t+Δt)=x(t) + l(t). The steps l(t) are given according to a distritubion χ whose moments are:

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\int \chi (z) dz = 1

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\int z\chi (z) dz = 0

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\int z^2\chi (z) dz = a^2

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For a particle to go from x' at t to x at t+Δt the step, l(t), must be x-x'. This happens with probability χ(x-x') times the probability density ρ(x', t) that it started at x'.

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\rho (x, t+\Delta t) = \int \rho (x', t)\chi (x-x') dx'

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\rho (x, t+\Delta t) = \int \rho (x-z, t)\chi (z) dz

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Assume ρ is broad and do a Taylor series.

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\rho (x, t+\Delta t) ≈ \int \left[\rho (x, t)-z\frac{\partial \rho}{\partial x}+z^2/2\frac{\partial^2 \rho}{\partial x^2} \right]\chi (z) dz

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Using the moments, we get:

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\rho (x, t+\Delta t) ≈ \rho (x, t)+1/2\frac{\partial^2 \rho}{\partial x^2}a^2

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Now assume ρ is slow and so changes little during this time step:

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\frac{\partial \rho}{\partial t}\Delta t ≈ \rho (x, t+\Delta t) - \rho (x, t)

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So we get:

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\frac{\partial \rho}{\partial t} = D\frac{\partial^2 \rho}{\partial x^2}

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where

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D = \frac{a^2}{2\Delta t}

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So far so good. But he says if we add an external force F whereby x(t+Δt)=x(t) + FγΔt + l(t), then we get

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\frac{\partial \rho}{\partial t} = -γF\frac{\partial \rho}{\partial x}+D\frac{\partial^2 \rho}{\partial x^2}

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But I am having trouble getting that result. It seems that the second derivative term is more like

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D+F^2γ^2Δt/2.

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Seeing as this is my first Taylor series expansion in over 20 years, I was wondering if one of you could help me out and tell me if I have made a mistake. It shouldn't take you very long.

Thanks!

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# Diffusion equation with an external force

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