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⇤ ← Revision 1 as of 2013-08-28 03:05:10
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| $$ {{F(x,(t+\delta t) - F(x,t)}\over{\delta t}} ~ = ~ D {{ F(x + \delta x, t) +F(x-\delta x, t) - 2*F(x,t)}\over{\delta x^2}} $$ | $$ {{F(x,t+\delta t) - F(x,t)}\over{\delta t}} ~ = ~ D {{ F(x + \delta x, t) +F(x - \delta x, t) - 2*F(x,t)}\over{\delta x^2}} $$ |
Diffusion
Diffusion turns curvatures in distributions into changes in time. The effect on a point on a curve is to take a little away from the point, and add a little from what is on both sides of it. This can be expressed as
{ { \partial F(x,t) } \over { \partial t } } ~ = ~ d { { \partial^2 F(x,t) } \over { \partial x^2 } }
Where F(x,t) is a function in space and time and d is the diffusion constant.
As a difference equation, this can be reformulated as:
{{F(x,t+\delta t) - F(x,t)}\over{\delta t}} ~ = ~ D {{ F(x + \delta x, t) +F(x - \delta x, t) - 2*F(x,t)}\over{\delta x^2}}
So, what is D? Let's figure it out for Rutherford diffusion.