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CHAPTER 9 Statistical Physics

- underpins thermodynamics, ideal gas (a classical

physics model), ensembles of molecules, solids,

liquids the universe - 9.1 Justification for its need !
- 9.2 Classical distribution functions as examples

of distributions of velocity and velocity2 in

ideal gas - 9.3 Equipartition Theorem
- 9.4 Maxwell Speed Distribution
- 9.5 Classical and Quantum Statistics
- 9.6 Black body radiation, Liquid Helium,

Bose-Einstein - condensates, Bose-Einstein

statistics, - 9.7 Fermi-Dirac Statistics

Ludwig Boltzmann, who spent much of his life

studying statistical mechanics, died in 1906 by

his own hand. Paul Ehrenfest, carrying on his

work, died similarly in 1933. Now it is our turn

to study statistical mechanics. Perhaps it will

be wise to approach the subject cautiously. -

David L. Goldstein (States of Matter, Mineola,

New York Dover, 1985)

First there was classical physics with a cause

(or causes)

Newtons three force laws, first unification in

physics

Lagrange around 1790 and Hamilton around 1840

added significantly to the computational power of

Newtonian mechanics. Pierre-Simon de Laplace

(1749-1827) Made major contributions to the

theory of probability and well known clockwork

universe statement It should be possible in

principle to have perfect knowledge of the

universe. Such knowledge would come from

measuring at one time the position and velocities

of every particle of matter and then applying

Newtons law. As they are cause and effect

relations that work forwards and backwards in

time, perfect knowledge can be extended all the

way back to the beginning of the universe and all

the way forward to its end. So no uncertainty

principle allowed

then there was the realization that one does not

always need to know the cause (causes), can do

statistical analyses instead

- Typical problem, flipping of 100 coins,
- One can try to identify all physical condition

before the toss, model the toss itself, and then

predict how the coin will fall down - if all done correctly, one will be able to make a

prediction on how many heads or tails one will

obtain in a series of experiments - Statistics and probabilities would just predict

50 heads 50 tails by ignoring all of that

physics, - The more experimental trials, 100,000 coin

tosses, the better this prediction will be borne

out

Speed distribution of particles in an ideal gas

in equilibrium, instead of analyzing what each

individual particle is going to do, one derives a

distribution function, determines the density of

states, and then calculates the physical

properties of the system (always by the same

procedures)

ltKEgt ltp2gt/2m

There is one characteristic kinetic energy (or

speed) distribution for each value of T, so we

would like to have a function that gives these

distribution for all temperatures !!!

Path to statistical physics from classical to

quantum for bosons and fermions

- Benjamin Thompson (Count Rumford) 1753 1814
- Put forward the idea of heat as merely the

kinetic energy of individual particles in an

ideal gas, speculation for other substances. - James Prescott Joule 1818 1889
- Demonstrated the mechanical equivalent of heat,

so central concept of thermodynamics becomes

internal energy of systems (many many particles

at once)

Beyond first or second year college physics

James Clark Maxwell 1831 1879, Josiah Willard

Gibbs 1839 1903, Ludwig Boltzmann 1844 1906

(all believing in reality of atoms, tiny minority

at the time) Brought the mathematical theories

of probability and statistics to bear on the

physical thermodynamics problems of their

time. Showed that statistical distributions of

physical properties of an ideal gas (in

equilibrium a stationary state) can be used to

explain the observed classical macroscopic

phenomena (i.e. gas laws) Gibbs invents notation

for vector calculus, the form in which we use

Maxwells equations today Maxwells

electromagnetic theory succeeded his work on

statistical foundation of thermodynamics so he

was a genius twice over.

and then there came modern physics

9.2 Maxwell Velocity and Velocity2 Distribution

- internal energy in an ideal gas depends only on

the movements of the entities that make up that

gas. - Define a velocity distribution function .
- the probability of finding a

particle with velocity - between .
- where

is similar to the product of a wavefunction with

its complex conjugate (in 3D), from it we can

calculate expectation values (what is measured on

average) by the same integration procedure as in

previous chapters !!

Maxwell Velocity Distribution

- Maxwell proved that the velocity probability

distribution function is proportional to exp(-½

mv2 / kT), special form of exp(-E/kT) the

Maxwell-Boltzmann statistics distribution

function. - Therefore where C is a proportionality

factor and ß (kT)-1. k Boltzmann constant,

which we find everywhere in this field - Because v2 vx2 vy2 vz2 then
- Rewrite this as the product
- of three factors.

Is the product of the three functions gx, gy gz

which are just for one variable (1D) each

Maxwell Velocity Distribution

- g(vx) dvx is the probability that the x component

of a gas molecules velocity lies between vx and

vx dvx. - if we integrate g(vx) dvx over all of vx and

set it equal to 1, we get the normalization

factor - The mean value (expectation value) of vx

Full Widths at Half Maximum e-0.5 0.607 g(0)

That is similar to the expectation value of

momentum in the square wells

Maxwell Velocity2 Distribution

- The mean value of vx2, also an expectation value

that is a simple function of x

This is not zero because it is related to kinetic

energy, remember the expectation value of p2 was

also not zero

It relates the human invented energy scale (at

the individual particle level) to the absolute

temperature scale (a physical thing)

1.3806488(13)10-23 J?K-1

8.6173324(78)10-5 eV?K-1

gas constant R divided by Avogadros number NA

Maxwell Velocity2 Distribution

- The results for the x, y, and z velocity2

components are identical. - The mean translational kinetic energy of a

molecule - Equipartion of the kinetic energy in each of 3

dimension a particle may travel, in each degree

of freedom of its linear movement - this result can be generalized to the

equipartition theorem

9.3 Equipartition Theorem

- Equipartition Theorem
- For a system of particles (e.g. atoms or

molecules) in equilibrium a mean energy of ½ kT

per system member is associated with each

independent quadratic term in the energy of the

system member. - That can be movement in a direction, rotation

about an axis, vibration about an equilibrium

position, , 3D vibrations in a harmonic

oscillator - Each independent phase space coordinate
- degree of freedom

Equipartition Theorem

- In a monatomic ideal gas, each molecule has
- There are three degrees of freedom.
- Mean kinetic energy is 3(1/2 kT) 3/2 kT
- In a gas of N helium atoms, the total internal

energy is - CV 3/2 N k
- For the heat capacity for 1 mole
- The ideal gas constant R 8.31 J/K

As predicted, only 3 translational degrees of

freedom

2 more (rotational) degrees of freedom

2 more (vibrational) degrees of freedom plus

vibration, which also adds two times 1/2 kBT

discrepancies due to quantized vibrations, not

due to high particle density

We get excellent agreement for the noble gasses,

they are just single particles and well isolated

from other particles

Molar Heat Capacity

- The heat capacities of diatomic gases are

temperature dependent, indicating that the

different degrees of freedom are turned on at

different temperatures. - Example of H2

The Rigid Rotator Model

- For diatomic gases, consider the rigid rotator

model. - The molecule rotates about either the x or y

axis. - The corresponding rotational energies are ½ Ix?x2

and ½ Iy?y2. - There are five degrees of freedom, three

translational and two rotational. (I is

rotational moment of inertia)

Two more degrees of freedom, ½ Ix?x2 and ½ Ix?x2

Two more degrees of freedom, because there are

kinetic and potential energy, both are

quadratic (both have variables that appear

squared in a formula of energy is a degree of

freedom, ½ m (dr/dt)2 and ½ ? r2

Using the Equipartition Theorem

- In the quantum theory of the rigid rotator the

allowed energy levels are - From previous chapters, the mass of an atom is

largely confined to its nucleus - Iz is much smaller than Ix and Iy. Only rotations

about x and y are allowed at reasonable

temperatures. - Model of diatomic molecule, two atoms connected

to each other by a massless spring. - The vibrational kinetic energy is ½ m(dr/dt)2,

there is kinetic and potential energy ½ ? r2 in a

harmonic vibration, so two extra degrees of

freedom - There are seven degrees of freedom (three

translational, two rotational, and two

vibrational for a two-atom molecule in a gas).

not that simple

six degrees of freedom

according to classical physics, Cv should be 3 R

6/2 kBT NA for solids and independent of the

temperature

We will revisit this problem when we have learned

of quantum distributions, concept of phonons,

which are quasi-particle that are not restricted

by the Pauli exclusion principle

Maxwells speed (v) distribution

Slits have small widths, size of it defines dv (a

small speed segment of the speed distribution)

9.4 Maxwell Speed Distribution ?v?

- Maxwell velocity distribution
- Where
- lets turn this into a speed distribution.
- F(v) dv the probability of finding a particle

with speed - between v and v dv.
- One cannot derive F(v) dv (i.e. a distribution of

a scalar entity) simply from f(v) d3v (the

velocity distribution function, i.e. a

distribution of vectors and their components), we

need idea of phase space for this derivation

Maxwell Speed Distribution

- Idea of phase space, to count how many states

there are - Suppose some distribution of particles f(x, y, z)

exists in normal three-dimensional (x, y, z)

space. - The distance of the particles at the point (x, y,

z) to the origin is - the probability of finding a particle

between .

Maxwell Speed Distribution

- Radial distribution function F(r), of finding a

particle between r and r dr sure not equal to

f(x,y,z) as we want to go from coordinates to

length of the vector, a scalar - The volume of any spherical shell is 4pr2 dr.
- now replace the 3D space coordinates x, y, and z
- with the velocity space coordinates vx, vy, and

vz - Maxwell speed distribution
- It is only going to be valid in the classical

limit, as a few particles would have speeds in

excess of the speed of light.

note speed distribution function is different to

velocity distribution function, but both have the

same Maxwell-Boltzmann statistical factor

Maxwell Speed Distribution

- The most probable speed v, the mean speed ,

and the root-mean-square speed vrms are all

different.

Maxwell Speed Distribution

- Most probable speed (at the peak of the speed

distribution), simply plot the function, take the

derivative and set it zero, derive the

consequences - Average (mean) speed, will be an expectation

value that we calculate from on an integral on

the basis of the speed distribution function

average (mean) of the square of the speed, will

be an expectation value that we calculate from

another integral on the basis of the speed

distribution function

We define root mean square speed on its basis

which is of course associated with the mean

kinetic energy

We can also calculate the spread (standard

deviation) of the speed distribution function in

analogy to quantum mechanical spreads

Note that sv in proportional to

So now we understand the whole function, can make

calculations for all T

Straightforward turn speed distribution into

kinetic energy (internal energy of ideal gas)

distribution

So we recover the equipartition theorem for a

mono-atomic gas

Number of particles with energy in interval E and

E dE

9.5 Needs for Quantum Statistics

- If molecules, atoms, or subatomic particles are

fermions, i.e. most of matter, in the liquid or

solid state, the Pauli exclusion principle

prevents two particles with identical wave

functions from sharing the same space. The

spatial part of the wavefunction can be identical

for two particles in the same state, but the spin

part f the wavefunction has to be different to

fulfill the Pauli exclusion principle. - If the particles under consideration are

indistinguishable and Bosons, they are not

subject to the Pauli exclusion principle, i.e.

behave differently - There are only certain energy values allowed for

bound systems in quantum mechanics. - There is no restriction on particle energies in

classical physics.

Classical physics Distributions

- Boltzmann showed that the statistical factor

exp(-ßE) is a characteristic of any classical

system in equilibrium (in agreement with

Maxwells speed distribution) - quantities other than molecular speeds may

affect the energy of a given state (as we have

already seen for rotations, vibrations) - Maxwell-Boltzmann statistics for classical

system ß (kBT)-1 - The energy distribution for classical system
- n(E) dE the number of particles with energies

between E and E dE. - g(E) the density of states, is the number of

states available per unit energy range. - FMB gives the relative probability that an energy

state is occupied at a given temperature.

A is a normalization factor, problem specific

Classical / quantum distributions

- Characteristic of indistinguishability that makes

quantum statistics different from classical

statistics. - The possible configurations for distinguishable

particles in either of two (energy or anything

else) states - There are four possible states the system can be

in.

State 1 State 2

AB

A B

B A

AB

Quantum Distributions

- If the two particles are indistinguishable
- There are only three possible states of the

system. - Because there are two types of quantum mechanical

particles, two kinds of quantum distributions are

needed. - Fermions
- Particles with half-integer spins, obey the Pauli

principle. - Bosons
- Particles with zero or integer spins, do not obey

the Pauli principle.

State 1 State 2

XX

X X

XX

Multiply each state with its number of

microstates for distinguishable particles sum

it all up and you get the distribution of

classical physics particles Ignore all

microstates for indistinguishable particles sum

it all up, that would be the distribution for

bosons Ignore all microstates and states that

have more than one particle at the same energy

level, - sum it all up, that would be the

distribution of fermions Serway et al, chapter 10

for details Realize, there must be three

different distribution functions !!

Quantum Distributions

Classical and Quantum Distributions

For photons in cavity, Planck, A 1, a 0

E is quantized in units of h if part of a bound

system

Quantum Distributions

If all three normalization factors 1, just for

comparison

has to do with specific normalization factor

- The exact forms of normalization factors for the

distributions depend on the physical problem

being considered. - Because bosons do not obey the Pauli exclusion

principle, more bosons can fill lower energy

states (are actually attracted to do so) - All three graphs coincide at high energies the

classical limit. - Maxwell-Boltzmann statistics may be used in the

classical limit when particles are so far apart

that they are distinguishable, can be tracked by

their paths

When there are so many states that there is a

very low probability of occupation

also if the particles are heavy (macroscopic),

i.e. a bunch of classical physics particle,

Bohrs correspondence principle again

Anything to do with solids, when high probability

of occupancy of energy states, e.g. electrons in

a metal, which are fermions

anything to do with liquids, when high

probability of occupancy of energy

states Bose-Einstein condensate for at 2.17 K

superfluidity (explained later on)

https//www.youtube.com/watch?v2Z6UJbwxBZI

degeneracy of the first exited state in H atom

n l ml ms up ms down

2 0 0 1/2 -1/2

2 1 1 1/2 -1/2

2 1 0 1/2 -1/2

2 1 -1 1/2 -1/2

g functions, density of states, how many states

there are per unit energy value, in other words

the degeneracy if we talk about a hydrogen atom

g functions are problem specific !!

revisited

Einsteins assumptions in 1907, atoms vibrate

independently of each other he used

Maxwell-Boltzmann statistics because there are so

many possibly vibration states that only a few of

the available states will be occupied, (and the

other distribution functions were not known at

the time)

(starting from zero point energy, due to

uncertainty principle)

A. Einstein, "Die Plancksche Theorie der

Strahlung und die Theorie der spezifischen

Wärme", Annalen der Physik 22 (1907) 180190

i.e. at high temperatures is approaches the

classical value of 2 degrees of freedom with ½ kT

each times 3 vibration direction (Bohrs

correspondence principle once more)

To account for different bond strength, different

spring constants

hetero-polar bond in diamond much stronger than

metallic bond in lead and aluminum, so much

larger Einstein Temperature for diamond (1,320 K)

gtgt 50-100 K for typical metals

Peter Debye lifted the assumption that atoms

vibrate independably, similar statistics, Debye

temperature TD even better modeling with

phonons, which are pseudo-particle of the boson

type

Blackbody Radiation

- Blackbody Radiation
- Intensity of the emitted radiation is
- Use Bose-Einstein distribution because photons

are bosons with spin 1 (they have two

polarization states) - For a free particle in terms of momentum in a 3D

infinitely deep well - E pc hf so we need the equivalent of this

formulae in terms of momentum (KE p2 / 2m)

now our particles are measles

Phase space again

Density of states in cavity, we can assume the

cavity is a sphere, we could alternatively assume

it is any kind of shape that can be filled with

cubes

Bose-Einstein Statistics

- The number of allowed energy states within

radius r of a sphere is - Where 1/8 comes from the restriction to positive

values of ni and 2 comes from the fact that there

are two possible photon polarizations. - Resolve Energy equation for r, and substitute

into the above equation for Nr - Then differentiate to get the density of states

g(E) is - Multiply the Bose-Einstein factor in

For photons, the normalization factor is 1, they

are created and destroyed as needed

Bose-Einstein Statistics

- Convert from a number distribution to an energy

density distribution u(E). - For all photons in the range E to E dE
- Using E hf and dE (hc/?2) d?
- In the SI system, multiplying by c/4 is required.

and world wide fame for Satyendra Nath Bose 1894

1974 !

Liquid Helium

- Has the lowest boiling point of any element (4.2

K at 1 atmosphere pressure) and has no solid

phase at normal pressure. - The density of liquid helium as a function of

temperature.

Liquid Helium

- The specific heat of liquid helium as a function

of temperature - The temperature at about 2.17 K is referred to as

the critical temperature (Tc), transition

temperature, or lambda point. - As the temperature is reduced from 4.2 K toward

the lambda point, the liquid boils vigorously. At

2.17 K the boiling suddenly stops. - What happens at 2.17 K is a transition from the

normal phase to the superfluid phase.

Thermal conductivity goes to infinity at lambda

point, so no hot bubbles can form while the

liquid is boiling,

Liquid Helium

- The rate of flow increases dramatically as the

temperature is reduced because the superfluid has

an extremely low viscosity. - Creeping film formed when the viscosity is very

low and some helium condenses from the gas phase

to the glass of some beaker.

Liquid Helium

- Liquid helium below the lambda point is part

superfluid and part normal. - As the temperature approaches absolute zero, the

superfluid approaches 100 superfluid. - The fraction of helium atoms in the superfluid

state - Superfluid liquid helium is referred to as a

Bose-Einstein condensation. - not subject to the Pauli exclusion principle

because (the most common helium atoms are bosons - all particles are in the same quantum state

https//www.youtube.com/watch?v2Z6UJbwxBZI

9.7 Fermi-Dirac Statistics

- EF is called the Fermi energy.
- When E EF, the exponential term is 1.
- FFD ½
- In the limit as T ? 0,
- At T 0, fermions occupy the lowest energy

levels. - Near T 0, there is no chance that thermal

agitation will kick a fermion to an energy

greater than EF.

Fermi-Dirac Statistics

T gt 0

- T 0

- As the temperature increases from T 0, the

Fermi-Dirac factor smears out. - Fermi temperature, defined as TF EF / k.

.

T gtgt TF

T TF

- When T gtgt TF, FFD approaches a decaying

exponential of the Maxwell Boltzmann statistics.

At room temperature, only tiny amount of fermions

are in the region around EF,i.e. can contribute

to elecric current,

Classical Theory of Electrical Conduction

- Paul Drude (1900) showed on the basis of the idea

of a free electron gas inside a metal that the

current in a conductor should be linearly

proportional to the applied electric field, that

would be consistent with Ohms law. - His prediction for electrical conductivity
- Mean free path is .
- Drude electrical conductivity

Classical Theory of Electrical Conduction

From Maxwells speed distribution

- According to the Drude model, the conductivity

should be proportional to T-1/2. - But for most metals is very nearly proportional

to T-1 !! - This is not consistent with experimental results.

- l and t make only sense for a realistic

microscopic model, so whole approach abandoned,

but free electron gas idea kept, just a different

kind of gas

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All condensed matter (liquids and solids)

problems are statistical quantum mechanics

problems !! Quantum condensed matter physics

problems are typically low temperature problems

Ideal gasses can be modeled classically,

because they have very low matter densities

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Quantum Theory of Electrical Conduction

- The allowed energies for electrons are
- The parameter r is the radius of a sphere in

phase space. - The volume is (4/3)pr 3.
- The exact number of states upto radius r is

.

Quantum Theory of Electrical Conduction

- Rewrite as a function of E
- At T 0, the Fermi energy is the energy of the

highest occupied level. - Total of electrons
- Solve for EF
- The density of states with respect to energy in

terms of EF

Quantum Theory of Electrical Conduction

- At T 0,
- The mean electronic energy
- Internal energy of the system
- Only those electrons within about kT of EF will

be able to absorb thermal energy and jump to a

higher state. Therefore the fraction of electrons

capable of participating in this thermal process

is on the order of kT / EF.

Quantum Theory of Electrical Conduction

- In general,
- Where a is a constant gt 1.
- The exact number of electrons depends on

temperature. - Heat capacity is
- Molar heat capacity is

Quantum Theory of Electrical Conduction

- Arnold Sommerfeld used correct distribution n(E)

at room temperature and found a value for a of p2

/ 4. - With the value TF 80,000 K for copper, we

obtain cV 0.02R, which is consistent with the

experimental value! Quantum theory has proved to

be a success. - Replace mean speed in Eq (9,37) by Fermi

speed uF defined from EF ½ uF2. - Conducting electrons are loosely bound to their

atoms. - these electrons must be at the high energy

level. - at room temperature the highest energy level is

close to the Fermi energy. - We should use

Quantum Theory of Electrical Conduction

- Drude thought that the mean free path could be no

more than several tenths of a nanometer, but it

was longer than his estimation. - Einstein calculated the value of l to be on the

order of 40 nm in copper at room temperature. - The conductivity is
- Sequence of proportions.

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- Rewrite Maxwell speed distribution in terms of

energy. - For a monatomic gas the energy is all

translational kinetic energy. - where

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