diffusion

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Diffusion on a lattice

Consider particles restricted to moving on an $d\,$ -dimensional lattice with spacing $a\,$ , such that during a time interval $\Delta t\,$ a particle may move to any adjacent lattice point with probability $p\, \Delta t\,$ (but not diagonally, for simplicity). Let the number of particles at a point $\mathbf{x}\,$ be given by $n(\mathbf{x})\,$ . In one dimension,

$n(x, t+\Delta t) = n(x,t) -2 p \Delta t n(x) + p \Delta t \left[ n(x+a,t) + n(x-a,t) \right]\,$ .

Let us represent $n(x,t)\,$ by a continuous function that takes on the exact values on the lattice points. Then

$n(x, t+\Delta t)\,$

$= n(x,t) + 2 p \Delta t \left[ a\frac{\partial n(x,t)}{\partial x} + \frac{a^2}{2}\frac{\partial^2 n(x,t)}{\partial x^2} -a\frac{\partial n(x,t)}{\partial x} + \frac{a^2}{2}\frac{\partial^2 n(x,t)}{\partial x^2} + O(a^3)\right]\,$ .