Cossey L. Introduction to Mathematical Physics

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Definition

Definition:

Let E be a topological vectorial space. The set of the continuous linear form on E is a

vectorila subspace of E

and is called the topological dual of E.

===Distributions===

Distributions{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.

Definition:

L. Schwartz distributions are linear functionals continuous on , thus the elements of the

dual of .

Definition:

A function is called locally summable if it is integrable in Lebesgue sense over any

bounded interval.

Definition:

To any locally summable function f, a distribution T

defined by:

can be associated.

Definition:

Dirac distribution,{Dirac distribution} noted δ is defined by:

Remark:

Physicist often uses the (incorrect!) integral notation:

to describe the action of Dirac distribution δ on a function φ.

Convolution of two functions f and g is the function h if exists defined by:

and is noted:

h = f * g.

Convolution product of two distributions S and T is (if exists) a distribution noted S * T

defined by:

Here are some results:

• convolution by δ is unity of convolution.

• convolution by is the derivation.

• convolution by δ

(m)

is the derivation of order m.

• convolution by δ(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:

Definition:

Let f(x) be a complex valuated function{Fourier transform} of the real variable x. Fourier

transform of f(x) is the complex valuated function

\sigma</math> defined by:

if it exists.

A sufficient condition for the Fourier transform to exist is that f(x) is summable. The

Fourier transform can be inverted: if

then

Here are some useful formulas:

Let us now generalize the notion of Fourier transform to distributions. The Fourier

transform of a distribution can not be defined by

Indeed, if

, then and the second member of previous equality does not

exist.

Definition:

Space

is the space of fast decreasing functions. More precisely, if

• its m − derivative φ (x)

(m)

exists for any positive integer m.

• for all positive or zero integers p and m, x φ (x)

p (m)

is bounded.

Definition:

Fourier transform of a tempered distribution T is the distribution defined by

The Fourier transform of the Dirac distribution is one:

Distributions{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{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 Ω of "results" ω. An event is an element of B, that is a set of

ω'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

. A random variable X

is an application from Ω into R (or C). For instance one can associate to each result ω 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.

Definition:

The density probability function f(x) is such that:

It satisfies:

Example:

The density probability function of a Bernoulli process is:

f(x) = pδ(x) + qδ(x − 1)

Moments of the partition function

Often, a function f(x) is described by its moments:

Definition:

The p

moment of function f(x) is the integral

Definition:

The mean of the random variable or mathematical expectation is moment m

Definition:

Variance v is the second order moment:

The square root of variance is called ecart-type and is noted σ.

Generating function

Definition:

The generatrice function{generating function} of probability density f is the Fourier

transform off.

Example:

For the Bernouilli distribution:

The property of Fourier transform:

implies that:

Sum of random variables

Theorem:

The density probability f

associated to the sum x = x + x

1 2

of two {\bf

independent} random variables is the convolution product of the

probability

densities and .

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 with N > n is n is, taking into account

all the possible cases:

This can be used to show the probability density associated to a binomial law. Using the

Fourier counterpart of previous theorem:

Let us state the central limit theorem.

Theorem:

The convolution product of a great number of functions tends\footnote{ The notion of

limit used here is not explicited, because this result will not be further used in this book. }

to a Gaussian.

{central limit theorem}

Proof:

Let us give a quick and dirty proof of this theorem. Assume that has the following

Taylor expansion around zero:

and that the moments m

with are zero. then using the definition of moments:

This implies using m = 1

that:

A Taylor expansion yields to

Finally, inverting the Fourier transform:

Differentials and derivatives

Definitions

Definition:

Let E and F two normed vectorial spaces on R or C.and f a mao defined on an open U of

E into F. f is said differentiable at a point x

of U if there esists a continous linear

application g from E into F such that

is negligible

with respect to .

The notion of derivative is less general and is usually defined for function for a part of R

to a vectorial space as follows:

Definition:

Let I be a interval of R different from a point and E a vectorial space normed on R. An

application f from I to E admits a derivative at the point x of I if the ratio:

admits a limit as h tends to zero. This limit is then called derivative of f at point x and is

noted .

We will however see in this appendix some genearalization of derivatives.

Derivatives in the distribution's sense

Definition

Derivative{derivative in the distribution sense} in the usual function sense is not defined

for non continuous functions. Distribution theory allows in particular to generalize the

classical derivative notion to non continuous functions.

Definition:

Derivative of a distribution T is distribution T' defined by:

Definition:

Let f be a summable function. Assume that f is discontinuous at N points a

, and let us

note

the jump of f at a

. Assume that f' is locally summable and

almost everywhere defined. It defines a distribution T

. Derivative (T)'

of the distribution

associated to f is:

One says that the derivative in the distribution sense is equal to the derivative without

precaution augmented by the Dirac distribution multiplied by the jump of f. It can be

noted:

===Case of distributions of several variables===

Using derivatives without precautions, the action of differential operators in the

distribution sense can be written, in the case where the functions on which they are acting

are discontinuous on a surface S:

grad f = { grad f} + nσ δ

f S

div a = { div f} + nσ δ

a S

where f is a scalar function, a a vectorial function, σ represents the jump of a or f through

surface S and δ

, is the surfacic Dirac distribution. Those formulas allow to show the

Green function introduced for tensors. The geometrical implications of the differential

operators are considered at next appendix chaptens

Example: Electromagnetism The fundamental laws of electromagnetism are the Maxwell

equations:{passage relation}

div D = ρ

div B = 0

are also true in the distribution sense. In books on electromagnetism, a chapter is

classically devoted to the study of the boundary conditions and passage conditions. The

using of distributions allows to treat this case as a particular case of the general equations.

Consider for instance, a charge distribution defined by:

ρ = ρ + ρ

v s

where ρ

is a volumic charge and ρ

a surfacic charge, and a current distribution defined

by:

j = j + j

v s

where j

is a volumic current, and j

a surfacic current. Using the formulas of section

secdisplu, one obtains the following passage relations:

where the coefficients of the Delta surfacic distribution δ

have been identified

Example: Electrical circuits

As Maxwell equations are true in the distribution sense previous example), the equation

of electricity are also true in the distribution sense. Distribution theory allows to justify

some affirmations sometimes not justified in electricity courses. Consider equation:

This equation implies that even if U is not continuous, i does. Indeed , if i is not

continuous, derivative would create a Dirac distribution in the second member.

Consider equation:

This equation implies that q(t) is continuous even if i is discontinuous.

Example: Fluid mechanics

Conservation laws are true at the distribution sense. Using distribution derivatives, so

called "discontinuity" relations can be obtained immediately

==Differentiation of Stochastic processes==

When one speaks of stochastic{stochastic process} processes , one adds the time notion.

Taking again the example of the dices, if we repeat the experiment N times, then the

number of possible results is Ω' = 6

(the size of the set Ω grows exponentialy with N).

We can define using this Ω' a probability P'. So, from the first random variable X, we can

define another random variable X

Definition: