Functional Derivative - Using The Delta Function As A Test Function

Using The Delta Function As A Test Function

The definition given above is based on a relationship that holds for all test functions f, so one might think that it should hold also when f is chosen to be a specific function such as the delta function. However, the latter is not a valid test function.

In the definition, the functional derivative describes how the functional changes as a result of a small change in the entire function . The particular form of the change in is not specified, but it should stretch over the whole interval on which is defined. Employing the particular form of the perturbation given by the delta function has the meaning that is varied only in the point . Except for this point, there is no variation in .

Often a physicist wants to know how one quantity, say the electric potential at position, is affected by changing another quantity, say the density of electric charge at position . The potential at a given position is a functional of the density, that is, given a particular density function and a point in space, one can compute a number which represents the potential of that point in space due to the specified density function. Since we are interested in how this number varies across all points in space, we treat the potential as a function of . To wit,

That is, for each, the potential is a functional of . Applying the definition of functional derivative,

$\begin{align} \left\langle \frac{\delta F}{\delta \rho(r')}, f(r') \right\rangle & {} = \frac{d}{d\varepsilon} \left. \frac{1}{4\pi\epsilon_0} \int \frac{\rho(r') + \varepsilon f(r')}{|r-r'|} \mathrm{d}r' \right|_{\varepsilon=0} \\ & {} = \frac{1}{4\pi\epsilon_0} \int \frac{f(r')}{|r-r'|} \mathrm{d}r' \\ & {} = \left\langle \frac{1}{4\pi\epsilon_0|r-r'|}, f(r') \right\rangle. \end{align}$

So,

$\frac{\delta V(r)}{\delta \rho(r')} = \frac{1}{4\pi\epsilon_0|r-r'|}.$

Now we can evaluate the functional derivative at and to see how the potential at is changed due to a small variation in the density at, but in general the unevaluated form is probably more useful.

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