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Good morning,

I have a question concerning the following terminologies : potential and energy.

Let $\mu$ be a finite Borel measure on $\mathbb{C}$ with compact support. We define its potential as the function $p_{\mu} \colon \mathbb{C}\to [-\infty,\infty)$ : $$p_{\mu}(z) = \int_{\mathbb{C}}\log|z-w|d\mu(w)$$ for $z\in\mathbb{C}.$

Then, we define the energy $I(\mu)$ by $$I(\mu) = \int_{\mathbb{C}}\int_{\mathbb{C}} \log|z-w|d\mu(z)d\mu(w).$$

My question : what is the meaning of these terminologies? Does anyone have a reference for these things?

I don't know much physics, so if someone answers this question with physics knowledge, please try to use very elementary examples in physics.

Thanks in advance,

Duc Anh

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2 Answers 2

up vote 3 down vote accepted

See, for example, this small book:

J.Wermer "Potential theory" (Lecture Notes in Mathematics, 408, 1974)

or any other book in mathematical physics concerning electrostatic or gravity theory.

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thank you very much, it seems to be an interesting book. –  Đức Anh Sep 18 '12 at 9:25

The three-dimensional analogues of these formulas may look more familiar:

$$ V(x)=\int\frac{\rho(x')}{|x-x'|}\mathrm dx'\;, $$

$$ E[\mu]=\int V(x)\rho(x)\mathrm dx=\iint\frac{\rho(x')\rho(x)}{|x-x'|}\mathrm dx'\mathrm dx\;, $$

where $\rho$ is a charge distribution. (This is actually twice the energy since the double integral double-counts the interactions; I wrote it like this in analogy to your expression.)

In three dimensions, a unit point charge at $x'$ generates a potential $1/|x-x'|$, and this is integrated with the charge density once to obtain the potential generated by the charge density, and again to obtain the interaction energy of the charge density with itself.

The point-charge potential $1/|x-x'|$ is the Green's function of Poisson's equation in three dimensions, that is, $\Delta(1/|x-x'|)=\delta(x-x')$, and the corresponding function in two dimensions is $\log|x-x'|$.

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thank you very much –  Đức Anh Sep 18 '12 at 9:24

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