Introduction to Mathematical Physics/Dual of a topological space

Definition

Distributions

Distributions\index{distribution} allow to describe in an elegant and synthetic way lots of physical phenomena. They allow to describe charge "distributions" in electrostatic (like point charge, dipole charge). They also allow to genereralize the derivation notion to functions that are not continuous.

Convolution of two functions $f$ and $g$ is the function $h$ if exists defined by: Template:IMP/eq and is noted: Template:IMP/eq Convolution product of two distributions $S$ and $T$ is (if exists) a distribution noted $S*T$ defined by: Template:IMP/eq Here are some results:

• convolution by $\delta$ is unity of convolution.
• convolution by ${\delta }^{\prime }$ is the derivation.
• convolution by ${\delta }^{(m)}$ is the derivation of order $m$.
• convolution by $\delta (x-a)$ is the translation of $a$.

The notion of Fourier transform of functions can be extended to distributions. Let us first recall the definition of the Fourier transform of a function:

Template:IMP/defn A sufficient condition for the Fourier transform to exist is that $f(x)$ is summable. The Fourier transform can be inverted: if Template:IMP/eq then Template:IMP/eq Here are some useful formulas: Template:IMP/eq Template:IMP/eq Template:IMP/eq Template:IMP/eq Template:IMP/eq Template:IMP/eq

Let us now generalize the notion of Fourier transform to distributions. The Fourier transform of a distribution can not be defined by Template:IMP/eq Indeed, if $\varphi \in 𝒟$, then $\varphi \notin 𝒟$ and the second member of previous equality does not exist. Template:IMP/defn Template:IMP/defn The Fourier transform of the Dirac distribution is one: Template:IMP/eq

Distributions\index{distribution} allow to describe in an elegant and synthetic way lots of physical phenomena. They allow to describe charge "distributions" in electrostatic (like point charge, dipole charge). They also allow to genereralize the derivation notion to functions that are not continuous.

Statistical description

Random variables

Distribution theory generalizes the function notion to describe\index{random variable} physical objects very common in physics (point charge, discontinuity surfaces,\dots). A random variable describes also very common objects of physics. As we will see, distributions can help to describe random variables. At section secstoch, we will introduce stochastic processes which are the mathemarical characteristics of being nowhere differentiable.

Let $B$ a tribe of parts of a set $\mathrm{\Omega }$ of "results" $\omega$. An event is an element of $B$, that is a set of $\omega$'s. A probability $P$ is a positive measure of tribe $B$. The faces of a dice numbered from 0 to 6 can be considered as the results of a set $\mathrm{\Omega }=\{{\omega }_1,\dots ,{\omega }_6\}$. A random variable $X$ is an application from $\mathrm{\Omega }$ into $R$ (or $C$). For instance one can associate to each result $\omega$ of the de experiment a number equal to the number written on the face. This number is a random variable.

Probability density

Distribution theory provides the right framework to describe statistical "distributions". Let $X$ be a random variable that takes values in $R$. Template:IMP/defn It satisfies: $\int f(x)dx=1$ Template:IMP/exmp

Moments of the partition function

Often, a function $f(x)$ is described by its moments: Template:IMP/defn Template:IMP/defn Template:IMP/defn

Generating function

Template:IMP/defn Template:IMP/exmp The property of Fourier transform: Template:IMP/eq implies that: Template:IMP/eq

Sum of random variables

Template:IMP/thm We do not provide here a proof of this theorem, but the reader can on the following example understand how convolution appears. The probability that the sum of two random variables that can take values in $i,\dots ,N$ with $N>n$ is $n$ is, taking into account all the possible cases: Template:IMP/eq This can be used to show the probability density associated to a binomial law. Using the Fourier counterpart of previous theorem:

$\begin{array}{ccc}\hat{f}(\nu )& =& (p+q{e}^{-2i\pi \nu }{)}^n\\ & =& \sum _{k=0}^n{C}_n^k{p}^{n-k}{q}^k{e}^{-2i\pi \nu k}\end{array}$

So Template:IMP/eq Let us state the central limit theorem. Template:IMP/thm Template:IMP/pf