Cossey L. Introduction to Mathematical Physics

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N body problems

At chapter chapproncorps, N body problem was treated in the quantum mechanics

framework. In this chapter, the same problem is tackled using statistical physics. The

number N of body in interaction is assumed here to be large, of the order of Avogadro

number. Such types of N body problems can be classified as follows:

• Particles are undiscernable. System is then typically a gas. In a classical approach,

partition function has to be described using a corrective factor section

secdistclassi). Partition function can be factorized in two cases: particles are

independent (one speaks about perfect gases){perfect gas} Interactions are taken

into account, but in the frame of a {\bf mean field approximation}{mean field}.

This allows to considerer particles as if they were independent van der Waals

model at section secvanderwaals) In a quantum mechanics approach, Pauli

principle can be included in the most natural way. The suitable description is the

grand-canonical description: number of particles is supposed to fluctuate around a

mean value. The Lagrange multiplier associated to the particles number variable

is the chemical potential µ. Several physical systems can be described by quantum

perfect gases (that is a gas where interactions between particles are neglected): a

fermions gas can modelize a semi--conductor. A boson gas can modelize helium

and described its properties at low temperature. If bosons are photons (their

chemical potential is zero), the black body radiation can be described.

• Particles are discernable. This is typically the case of particles on a lattice. Such

systems are used to describe for instance magnetic properties of solids. Taking

into account the interactions between particles, phase transitions like

paramagnetic--ferromagnetic transition can be described\footnote{ A mean field

approximation allows to factorize the partition function. Paramagnetic--

ferromagnetic transition is a second order transition: the two phases can coexist,

on the contrary to liquid vapour transition that is called first order transition.}.

Adsorption phenomenom can be modelized by a set of independent particles in

equilibrium with a particles reservoir (grand-canonical description).

Those models are described in detail in. In this chapter, we recall the most important

properties of some of them.

Thermodynamical perfect gas

In this section, a perfect gas model is presented: all the particles are independent, without

any interaction.

Remark:

A perfect gas corresponds to the case where the kinetic energy of the particles is large

with respect to the typical interaction energy. Nevertheless, collisions between particles

that can be neglected are necessary for the thermodynamical equilibrium to establish.

Classical approximation section secdistclassi) allows to replace the sum over the

quantum states by an integral of the exponential of the classical hamiltonian H(q ,p )

i i

. The

price to pay is just to take into account a proportionality factor

. Partition function z

associated to one particle is:

Partition function z is thus proportional to V :

z = A(β)V

Because particles are independent, partition function Z for the whole system can be

written as:

It is known that pressure (proportional to the Lagrange multiplier associated to the

internal variable "volume") is related to the natural logarithm of Z; more precisely if one

sets:

F = − k TlogZ

then

This last equation and the expression of Z leads to the famous perfect gas state equation:

pV = Nk T

Van der Waals gas

Van der Waals\footnote{Johannes Diderik van der Waals received the physics Nobel

price in 1910.} gas model{Van der Walls gas} is a model that also relies on classical

approximation, that is on the repartition function that can be written:

where the function is the hamiltonian of the system.

As for the perfect gas, integration over the p

's is immediate:

with

We can simplify the evaluation of previuos integral in neglecting correlations between

particles. A particle number i doesn't feel each particle but rather the average influence of

the particles cloud surrounding the considered particle; this approximation is called mean

field approximation {mean field}. It leads, as if the particles were actually independant to

a factorization of the repartition function. Interaction potential

becomes:

where U (r )

e i

is the "effective potential" that describes the mean interaction between

particle i and all the other particles. It depends only on position of particle i. Function Y

introduced at equation ---eqYint--- can be factorized:

In a mean field approximation framework, it is considered that particle distribution is

uniform in the volume. Effective potential has thus to traduce an mean attraction. Indeed,

it can be shown that at large distances two molecules are attracted, potential varying like

. This can be proofed using quantum mechanics but this is out of the frame of this

book. At small distances however, molecules repel themselves strongly. Effective

potential undergone by a particle at position r = 0 can thus be modelled by function:

Quantity Y is

Quantity bN represents the excluded volume of the particle. It is proportional to N

because N particles occupying each a volume b occupy a Nb. On another hand mean

potential U

felt by the test particle depends on ratio N / V. Usually, one sets:

Introducing

F = − k TlogZ

and using

one finally obtains Van der Waals state equation:

Van der Waals model allows to describe liquid--vapour phase transition. [[ Image:

When temperature is lower than critical temperature T

, the energy of the system for a

given volume V

with is not F (V )

v M

. Indeed, system evolves to a

state of lower energy figure

figvanapres) with energy:

The apparition of a local minimum of F corresponds to the apparition of two

phases.{phase transition}

Ising Model

In this section, an example of the calculation of a partition function is presented. The

Ising model

{Ising} is a model describing ferromagnetism{ferromagnetic}. A ferromagnetic material

is constituted by small microscopic domains having a small magnetic moment. The

orientation of those moments being random, the total magnetic moment is zero. However,

below a certain critical temperature T

, magnetic moments orient themselves along a

certain direction, and a non zero total magnetic moment is observed\footnote{ Ones says

that a phase transition occurs.{phase transition} Historically, two sorts of phase

transitions are distinguished

phase transition of first order (like liquid--vapor transition) whose

characteristics are:

1. Coexistence of the various phases.

2. Transition corresponds to a variation of entropy.

3. existence of metastable states.

second order phase transition (for instance the

ferromagnetic--paramagnetic transition) whose characteristics are:

1. symmetry breaking

2. the entropy S is a continuous function of temperature and of the order

parameter.

} . Ising model has been proposed to describe this phenomenom. It consists in describing

each microscopic domain by a moment S

(that can be considered as a spin){spin}, the

interaction between spins being described by the following hamiltonian (in the one

dimensional case):

partition function of the system is:

which can be written as:

It is assumed that S

can take only two values. Even if the one dimensional Ising model

does not exhibit a phase transition, we present here the calculation of the partition

function in two ways. represents the sum over all possible values of S

, it is thus, in

the same way an integral over a volume is the successive integral over each variable, the

successive sum over the S

's. Partition function Z can be written as:

with

f (S ,S ) = ch K + SS sh K

K i i + 1 i i + 1

We have:

Indeed:

Thus, integrating successively over each variable, one obtains:

IMP / label | eqZisiZ = 2 ( ch K)

n − 1 n − 1

This result can be obtained a powerful calculation method: the {\bf

renormalization group

method}

{renormalisation group} proposed by K. Wilson\footnote{Kenneth

Geddes Wilson received the physics Nobel price in 1982 for the

method of analysis introduced here.}.

Consider again the partition function:

where

f (S ,S ) = ch K + SS sh K

K i i + 1 i i + 1

Grouping terms by two yields to:

where

g(S ,S ,S ) = ( ch K + SS sh K)( ch K + S S sh K)

i i + 1 i + 2 i i + 1 i + 1 i + 2

This grouping is illustrated in figure figrenorm. [[ Image:

renorm

| center | frame |Sum over all possible spin values S ,S ,S

i i + 1 i + 2

. The

product

f (S ,S )f (S ,S )

K i i + 1 K i + 1 i + 2

is the sum over all possible values of spins S

and S

i + 2

of a

function f (S ,S )

K' i i + 2

deduced from f

by a simple change of the value of the parameter K

associated to function f

]] Calculation of sum over all possible values of S

i + 1

yields to:

Function can thus be written as a second function f (S ,S )

K' i i + 2

with

K' = Arcth ( th K).

Iterating the process, one obtains a sequence converging towards the partition function Z

defined by equation eqZisi.

Spin glasses

Assume that a spin glass system {spin glass} section{secglassyspin}) has the energy:

Values of variable S

are + 1 if the spin is up or - 1 if the spin is down. Coefficient J

is +

1 if spins i and j tend to be oriented in the same direction or - 1 if spins i and j tend to be

oriented in opposite directions (according to the random position of the atoms carrying

the spins). Energy is noted:

where J in H

denotes the J

distribution. Partitions function is:

where [s] is a spin configuration. We look for the mean over J

distributions of the

energy:

where P[j] is the probability density function of configurations [J], and where f

is:

f = − lnZ .

J J

This way to calculate means is not usual in statistical physics. Mean is done on the

"chilled" J variables, that is that they vary slowly with respect to the S

's. A more classical

mean would consist to

J</math>'s are then

"annealed" variables). Consider a system compound by n replicas{replica} of the

same system S

. Its partition function is simply:

Let f

be the mean over J defined by:

As:

we have:

Using

∑

P[J] = 1

and ln(1 + x) = x + O(x) one has:

By using this trick we have replaced a mean over lnZ by a mean over Z

; price to pay is

an analytic prolongation in zero. Calculations are then greatly simplified .

Calculation of the equilibrium state of a frustrated system can be made by simulated

annealing method .{simulated annealing} An numerical implementation can be done

using the Metropolis algorithm{Metropolis}. This method can be applied to the travelling

salesman problem

{travelling salesman problem}).

Quantum perfect gases

Introduction

Consider a quantum perfect gas, {\it i. e.}, a system of independent particle that have to

be described using quantum mechanics, {quantum perfect gas} in equilibrium with a

particle reservoir and with a thermostat. The description adopted is called "grand--

canonical". It can be shown that the solution of the entropy maximalization problem

(using Lagrange multipliers method) provides the occupation probability P

of a state l

characterized by an energy E

an a number of particle N

with

Assume that the independent particles that constitute the system are in

addition identical undiscernible. State l can then be defined by the

datum of the states λ of individual particles.

Let ε

be the energy of a particle in the state λ. The energy of the system in state l is then:

∑

where N

is the number of particles that are in a state of energy E

. This number is called

occupation number of state of energy{occupation number} E

figure ---figoccup---).

Thus:

∑

Partition function becomes:

Z = Π ξ

λ λ

with