Cossey L. Introduction to Mathematical Physics

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Solve this non linear problem (find solution ) assuming:

• B field is directed along direction z and so defines parallel direction and a

perpendicular direction (the plane perpendicular to the B field).

• speeds can be written with and .

• field can be written: .

• densities can be written . Plasma satisfies the quasi--neutrality

condition{quasi-neutrality} : .

• gases are considered perfect: p = n k T

a a B a

• T

, T

have the values they have at equilibrium.

Introduction

The first principle of thermodynamics section secpremierprinci) allows to bind the

internal energy variation to the internal strains power {strains}:

if the heat flow is assumed to be zero, the internal energy variation is:

dU = − P dt

This relation allows to bind mechanical strains (P

term) to system's thermodynamical

properties (dU term). When modelizing a system some "thermodynamical" variables X

are chosen. They can be scalars x, vectors x

, tensors x

, \dots Differential dU can be

naturally expressed using those thermodynamical variables X by using a relation that can

be symbolically written:

dU = FdX

where F is the conjugated\footnote{ This is the same duality relation noticed between

strains and speeds and their gradients when dealing with powers.}

thermodynamical variable of variable

X. In general it is looked for expressing F as a function of X .

Remark:

If X is a scalar x, the energy differential is:

dU = f.dx

If X is a vector x

, the energy differential is:

dU = f dx

i i

Si X is a tensor x

, the energy differential is:

dU = f dx

ij ij

Remark:

If a displacement x, is considered as thermodynamical variable, then the conjugated

variable f has the dimension of a force.

Remark:

One can go from a description using variable X as thermodynamical variable to a

description using the conjugated variable F of X as thermodynamical variables by using a

Legendre transformation section secmaxient and

The next step is, using physical arguments, to find an {\bf expression of the internal

energy U(X)} {internal energy} as a function of thermodynamical variables X. Relation

F(X) is obtained by differentiating U with respect to X, symbolically:

In this chapter several examples of this modelization approach are presented.

Electromagnetic energy

Introduction

At chapter chapelectromag, it has been postulated that the electromagnetic power given

to a volume is the outgoing flow of the Poynting vector. {Poynting vector} If currents are

zero, the energy density given to the system is:

dU = HdB + EdD

Multipolar distribution

It has been seen at section secenergemag that energy for a volumic charge distribution ρ

is {multipole}

where V is the electrical potential. Here are the energy expression for common charge

distributions:

• for a point charge q, potential energy is: U = qV(0).

• for a dipole {dipole} P

potential energy is: .

• for a quadripole Q

i,j

potential energy is:

Consider a physical system constituted by a set of point charges q

located at r

. Those

charges can be for instance the electrons of an atom or a molecule. let us place this

system in an external static electric field associated to an electrical potential U

. Using

linearity of Maxwell equations, potential U (r)

felt at position r is the sum of external

potential U (r)

and potential U (r)

created by the point charges. The expression of total

potential energy of the system is:

In an atom,{atom} term associated to V

is supposed to be dominant because of the low

small value of r − r

n m

. This term is used to compute atomic states. Second term is then

considered as a perturbation. Let us look for the expression of the second term

. For that, let us expand potential around r = 0 position:

where labels position vector of charge number n. This sum can be written as:

the reader recognizes energies associated to multipoles.

Remark: In quantum mechanics, passage laws from classical to quantum mechanics

allow to define tensorial operators chapter chapgroupes) associated to multipolar

momenta.

Field in matter

In vacuum electromagnetism, the following constitutive relation is exact:

D = ε E

Those relations are included in Maxwell equations. Internal electrical energy variation is:

dU = EdD

or, by using a Legendre transform and choosing the thermodynamical variable E:

dF = DdE

We propose to treat here the problem of the model

ization of the function D(E). In other

words, we look for the medium constitutive relation. This problem can be treated in two

different ways. The first way is to propose {\it a priori} a relation D(E) depending on the

physical phenomena to describe. For instance, experimental measurements show that D is

proportional to E. So the constitutive relation adopted is:

D = χE

Another point of view consist in starting from a m

icroscopic level, that is to modelize the

material as a charge distribution is vacuum. Maxwell equations in vacuum eqmaxwvideE

and eqmaxwvideB can then be used to get a macroscopic model. Let us illustrate the first

point of view by some examples:

Example:

If one impose a relation of the following type:

D = ε E

i ij j

then medium is called dielectric .{dielectric} The expression of the energy is:

F = F + ε E E

0 ij i j

Example:

In the linear response theory {linear response}, D

at time t is supposed to depend not

only on the values of E at the same time t, but also on values of E at times anteriors. This

dependence is assumed to be linear:

D (t) = ε *

i ij j

where * means time convolution.

Example:

To treat the optical

activity

The second point of view is now illustrated by the following two examples:

Example:

A simple model for the susceptibility: {susceptibility} An elementary ele

ctric dipole

located at r

can be modelized section secmodelcha) by a charge distribution div (pδ(r )

Consider a uniform distribution of N such dipoles in a volume V, dipoles being at position

. Function ρ that modelizes this charge distribution is:

ρ =

∑

div (p

δ(r

))

As the divergence operator is linear, it can also be written:

ρ = div

∑

δ(r

))

Consider the vector:

This vector P is called polarization vector{polarisation}. The evaluation of this vector P

is illustrated by figure figpolar.

Polar

Polarization vector at point r is the limit of the ratio of the sum of elementary dipolar

moments contained in the box dτ over the volume d\tau</math> as it tends towards zero.}

Maxwell--gauss equation in vacuum

div ε E = ρ

can be written as:

div (ε E − P) = 0

We thus have related the microscopic properties of the material (the p's) to the

macroscopic description of the material (by vector D = ε E − P

). We have now to provide

a microscopic model for p. Several models can be proposed. A material can be

constituted by small dipoles all oriented in the same direction. Other materials, like oil,

are constituted by molecules carrying a small dipole, their orientation being random when

there is no E field. But when there exist an non zero E field, those molecules tend to

orient their moment along the electric field lines. The mean P of the p

's given by

equation eqmoyP that is zero when E is zero (due to the random orientation of the

moments) becomes non zero in presence of a non zero E. A simple model can be

proposed without entering into the details of a quantum description. It consist in saying

that P is proportional to E:

P = χE

where χ is the polarisability of the medium. In this case relation:

D = ε E − P

becomes:

D = (ε + χ)E

Example: A second model of susceptibility: Consider the Vlasov equation equation

eqvlasov and reference

Generalized elasticity

Introduction

In this section, the concept of elastic energy is presented. {elasticity} The notion of

elastic energy allows to deduce easily "strains--deformations" relations.{strain--

deformation relation} So, in modelization of matter by virtual powers method {virtual

powers} a power P that is a functional of displacement is introduced. Consider in

particular case of a mass m attached to a spring of constant k.Deformation of the system

is referenced by the elongation x of the spring with respect to equilibrium. The virtual

work {virtual work} associated to a displacement dx is

δW = f.dx

Quantity f represents the constraint , here a force, and x is the deformation. If force f is

conservative, then it is known that the elementary work (provided by the exterior) is the

total differential of a potential energy function or internal energy U :

δW = − dU

In general, force f depends on the deformation. Relation f = f(x) is thus a constraint--

deformation relation .

The most natural way to find the strain-deformation relation is the following. One looks

for the expression of U as a function of the deformations using the physics of the the

problem and symmetries. In the particular case of an oscillator, the internal energy has to

depend only on the distance x to equilibrium position. If U admits an expansion at x = 0,

in the neighbourhood of the equilibrium position U can be approximated by:

U(x) = a + a x + a x + O(x )

0 1

2 2

As x = 0 is an equilibrium position, we have dU = 0 at x = 0. That implies that a

is zero.

Curve U(x) at the neighbourhood of equilibrium has thus a parabolic shape figure

figparabe

Parabe

In the neighbourhood of a stable equilibrium position x

, the intern energy function U, as

a function of the difference to equilibrium presents a parabolic profile.

dU = − fdx

the strain--deformation relation becomes:

Oscillators chains

Consider a unidimensional chain of N oscillators coupled by springs of constant k

. this

system is represented at figure figchaineosc. Each oscillator is referenced by its

difference position x

with respect to equilibrium position. A calculation using the

Newton's law of motion implies:

Chaineosc

A coupled oscillator chain is a toy example for studying elasticity.

A calculation using virtual powers principle would have consisted in affirming: The total

elastic potential energy is in general a function

x_i</math> to the equilibrium

positions. This differential is total since force is conservative\footnote{ This assumption

is the most difficult to prove in the theories on elasticity as it will be shown at next

section} . So, at equilibrium: {equilibrium} :

dU = 0

If U admits a Taylor expansion:

In this last equation, repeated index summing convention as been used. Defining the

differential of the intern energy as:

dU = f dx

i i

one obtains

Using expression of U provided by equation eqdevliUch yields to:

f = a x .

i ij j

But here, as the interaction occurs only between nearest neighbours, variables x

are not

the right thermodynamical variables. let us choose as thermodynamical variables the

variables ε

defined by:

ε = x − x .

i i i − 1

Differential of U becomes:

dU = F dε

i i

Assuming that U admits a Taylor expansion around the equilibrium position:

U = b + b ε + b ε ε + O(ε )

i i ij i j

and that dU = 0 at equilibrium, yields to:

F = b ε

i ij j

As the interaction occurs only between nearest neighbours:

so:

F = b ε + b ε

i ii i ii + 1 i + 1

This does correspond to the expression of the force applied to mass i :

F = − k(x − x ) − k(x − x )

i i i − 1 i + 1 i

if one sets k = − b = − b

ii ii + 1

Tridimensional elastic material

Consider a system S in a state S

which is a deformation from the state S

. Each particle

position is referenced by a vector a in the state S

and by the vector x in the state S

x = a + X

Vector X represents the deformation.

Remark:

Such a model allows to describe for instance fluids and solids.

Consider the case where X is always "small". Such an hypothesis is called small

perturbations hypothesis (SPH). The intern energy is looked as a function U(X).

Definition: The deformation tensor SPH is the symmetric part of the tensor gradient of X.

At section secpuisvirtu it has been seen that the power of the admissible intern strains for

the problem considered here is:

with

dU = − P dt

Tensor is called rate of deformation tensor. It is the symmetric part of tensor u

i,j

. It

can be shown that in the frame of SPH hypothesis, the rate of deformation tensor is

simply the time derivative of SPH deformation tensor:

Thus: