Excerpts of Some Statistical Mechanics

Lectures Found on the Web

Statistical Mechanicsin the

Canonical Ensemble

Outline of the Formalism

and

Some Examples(similar to parts of

Chs. 6 & 7in Reif’s book)

The Canonical Ensemble:Outline of the General Formalism

• This is THE CANONICAL DISTRIBUTION & gives “the probability that a system in contact with a heat bath at temperature T should be in a particular state”.

• r labels the states of the system. At low temperatures, only the lowest states have any chance of being occupied. As the temperature is raised, higher lying states become more and more likely to be occupied.

• In this case, in contact with the heat bath, all microstates are therefore not equally likely to be populated.

kT

U

r

r

eZ

p

1

r

kT

Ur

eZ

• Often, there are huge numbers of microstates that can all have the same energy. This is called DEGENERACY.

• In this case, the summations are over each individual energy level rather than sum over each microstate.

• The sum is now over each different energy Ur & g(Ur) is the number of states possessing the energy Ur. The probability is that of finding the system with energy Ur.

kT

U

rr

r

eUgZ

Up

)(1

)(

r

r

U

kT

U

r eUgZ )(

Canonical DistributionkT

U

r

r

eZ

p

1

r

kT

Ur

eZ

Entropy in the Canonical Ensemble• The system of interest is in equilibrium with a heat bath for which

the energy of the system fluctuates & the probability of finding any particular microstate is variable. How can the entropy be calculated for such a system? If the entropy S can be found, all thermodynamic variables can be calculated.

• The system is in a heat bath made up of (M-1) replica subsystems to the one of interest.

• Each subsystem may be in one of many microstates. The number of subsystems in the ith microstate is ni.

• The number of ways of arranging n1 systems of microstate 1, n2 systems of microstate 2, n3….is:

!

!

iin

MW

Entropy

• This is the general definition of entropy & holds even if the probabilities of each individual microstate are different.

i

ii ppkS ln

WkWW

kppkSW

iiii ln

1ln

1ln

1

Entropy in The Canonical Ensemble

• The general Definition of Entropy, in combination with The Canonical Distribution allows the calculation of the system thermodynamic properties:

i

ii ppkS ln

i

ii

iii Z

kT

UpkppkS lnln

kT

U

i

i

eZ

p

1

i

kT

Ui

eZ

ZkT

Up i

i lnln

ZkT

UpZkUp

TS

ii

iii lnln

1

Helmholtz Free Energy

• Ū Average value of the systemInternal Energy.

• (Ū - TS) Average value of theHelmholtz Free Energy, F.

• The Partition Function Z is much more than a mere normalizing factor. Z: Acts as a “Bridge” linking microscopic physics (quantum states) to the energy & so to all macroscopic properties of a system.

ZkT

US ln ZkTTSUF ln

• F is a state function. Now, the calculation of some thermodynamic properties of the system.

• Ignore the fact that Ū is an average & let U = Ū.• Use the definitions for various thermodynamic variables.

For an infinitesimal, quasistatic reversible change:

ZkTTSUF ln

TSUF SdTTdSdUdF

SdTTdSPdVdQdF R

SdTPdVSdTTdSPdVTdSdF

Helmholtz Free Energy

ZkTTSUF lnSdTPdVdF

• Using the properties of partial derivatives gives:

• So, the energies of the microstates of the system are linked to thermodynamic variables such as pressure & entropy.

PV

F

T

TT V

ZTk

V

FP

ln

ST

F

V

VV T

ZTk

T

FS

ln

Various Measurable Parameters:2nd Derivatives of the Helmholtz Free Energy F

• Elastic Moduli are the stress/strain or force/unit area divided by the fractional deformation:

• Heat Capacity at constant volume:TT V

FV

V

PVK

2

2

VVV

V T

FT

T

ST

T

QC

2

2

Mean Internal Energy• Ū Thermal Average of the system Internal

Energy. The actual internal energy fluctuates because the system is interacting with the heat bath.

• How large are the fluctuations? Are they important?

VT

ZTkTZkTTSFU

)ln(

ln

T

ZkT

T

ZTZkTZkTU

V

lnln

lnln 2

Fluctuations in Internal Energy• A measure of the departure from the mean is the

standard deviation, as it is in any statistical theory.

2222)( UUUUU

Vr

kT

Ui

kT

U

i

V

V r

i

e

eU

TT

UC

22

2

1UU

kTe

eU

TC

Vr

kT

Ui

kT

U

i

V r

i

VCkTUUU 2222)(

2222)( UUUUU

The VarianceThe relative fluctuation (U/ Ū)

gives the most useful information.

• Ū & CV are extensive properties proportional to the size of the system ~ N ( Number of particles in the system).

U

CkT

U

U V

2

NU

U 1

• For Typical Macroscopic Systems with ~1023 particles, the fluctuations (U/ Ū) ~ 10-11

• So, the fluctuations are tiny, which means that U & Ū can be considered identical for all practical purposes.

• Based on this, it is clear that Macroscopic Systems interacting with a heat bath effectively have their energy determined by that interaction.

• Similar relationships can be found for other relative fluctuations of properties of macroscopic systems.

NU

U 1

Summary – Statistical Mechanics• Microstate – The state of a system defined microscopically

– a complete description on the atomic scale.• Macrostate – The state of a stystem of macroscopic size

specified by a few macroscopically observable quantities only.• Statistical Weight (W or ) of a macrostate – is the

number of microstates compising the macrostate.• Postulate of equal a priori probabilities – “for an

isolated system in a definite macrostate, the W microstates comprising this macrostate occur with equal probability.

• Equilibrium Postulate – “For an isolated macroscopic system, defined by U,V,N (which are fixed) and variable parameters , equilibrium corresponds to those values of for which the statistical weight W(U,V,N, ) attains its maximum.”

Boltzmann Definition of Entropy

Definition of Temperature

Definition of Pressure

General Definition of Entropy

),,,(ln),,,( NVUWkNVUS

NUV

NVUSTP

,

),,(

i

ii ppkS ln

NVU

NVUS

T ,

),,(1

Canonical Ensemble Distribution• pi Probability that the system at temperature T

is in the state i with energy Ui:

• Partition Function Sum over All Microstates:

• Mean Energy:

• Helmholtz Free Energy:

kT

U

i

i

eZ

p

1

T

ZkTU

ln2

i

kT

U i

eZ

TSUF

Application

A Simple Modelof Paramagnetism

Section 6.3 (Spin = ½)Section 7.8 (Spin = S)

Paramagnetic Materials• This is a simple model to develop our

understanding of Statistical Mechanics but it proves to be very significant.

• Paramagnets contain atoms which have magnetic dipole moments (). These do not interact with each other but can respond to an applied external magnetic (B) field.

• The dipoles can be crudely thought of as independent (atomic) bar magnets arranged on a crystal lattice.

• The dipoles can be crudely thought of as independent (atomic) bar magnets arranged on a crystal lattice.

• Crude Picture: • In a B field each dipole can exist in one of two

states – aligned with the field (spin up) or anti aligned (spin down).

• Spin up dipoles have an energy -B, spin down +B.• We want to find out how the magnetization of the

material depends on the temperature and the applied field.

• As all the dipoles are independent of each other we really only need to look at the average properties of one dipole. We can use all the other dipoles as the heat bath – Canonical Ensemble.

• We have the two possible microstates and energies already – get Partition Function Z1 for our single dipole.

xkT

BeeeZ kT

B

kT

B

i

kT

Ui

cosh2cosh2

kT

Bx

• We can now calculate the probability of spin up versus spin down.

kT

U

i

i

eZ

p

1 xe

xp

cosh2

1 xex

p

cosh2

1

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Pro

ba

bilit

y

X=B/kT

Spin aligned (U = -B) Spin antialigned (U = +B)

• We can now calculate the mean magnetic moment of our individual dipole.

• So the mean energy of an individual dipole is:

xex

p

cosh

1 xex

p

cosh

1

xx

eex

pp xx sinh2cosh2

)(cosh2

xtanh

xBU tanh

• We have the values necessary for the individual dipole and because our dipoles do not interact all other dipoles must behave similarly.

• A solid of N dipoles therefore has an energy:

• And a mean magnetic moment (or magnetization) in the direction of the applied field of:

• Note that U = -MB

xNNM tanh

xNBUNU tanh

• The magnetization or the magnetic moment L per unit volume:

• In the limit of a tanh x x weak field or high temperature x<<1 and so:

V

xN

V

ML

tanh

BVkT

N

V

NxL

2

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Spin Up Spin Down Magnetisation

X=b/kT

Pro

bab

ility

0.0

0.2

0.4

0.6

0.8

1.0M

agn

etisatio

n / N/V

Reif, Figure 6-3-1

“Saturation Magnetization”

“Curie’s Law” of Paramagnetism

M0 χH, χ (N0μ2)/(kT)χ “Curie Susceptibility”

Curie’s “Law”• The susceptibility is the magnetization per

applied field intensity which for small magnetizations is given by H = B/0.

• This is Curie’s “Law”. It holds very well for paramagnetic materials with weakly interacting dipoles.

• It works so well that it can be used for temperature calibration. For example Cerium Magnesium Nitrate obeys Curie’s “Law” to 0.01K!

VkT

N

H

L 0

2 T

1

vs. T plot •1/ = T/C gives a straight line of gradient C-1 and intercept zero

• T = C gives a straight line parallel to the x-axis at a constant value of

T, thus showing the temperature independence of the magnetic moment.

Curie “Law”•Because single electrons are magnets, if you place them in a

magnetic field, they’ll align with the field. However the energy difference between aligned with field and against field is << thermal energy at room temperature. So, there are randomorientations, with equal populations of alignment with/against field.

•As you lower T, energy difference becomes more important & the population changes, with more aligned anti-parallel to the field.

• To explain this behavior, Curie invented a parameter – called the “Magnetic Susceptibility”, χ, – which is a measure of how attracted a sample is to a magnetic field. This is normallymeasured as an apparent mass increase. As more electrons alignanti-parallel to the field at low temperature, χ increases. In fact,

χ is inversely proportional to the temperature. This is the

Curie “Law”: (1/χ) = CTC = “The Curie Constant”

Magnetic Heat Capacity

• The paramagnetic solid has an energy that depends on temperature It therefore must have a magnetic heat capacity.

• Experimentally one measures the heat capacity at constant magnetic field intensity H.

• The paramagnetic compound is weakly magnetic, soB = 0 H

so B is also constant.

kT

BNBxNBUNU

tanhtanh

kT

B

TNB

T

U

T

dQC

HH

H

tanh

Magnetic Heat Capacity

kT

B

kT

BNk

kT

B

TNBCH

2

2

sechtanh

0 1 2 3 4 5

0.0

0.2

0.4

CH /

Nk

1/X = kT/B

Magnetic Heat Capacity

This is also called The Schottky Heat Capacity

kT

B

kT

BNkCH

2

2

sech

0 1 2 3 4 5

0.0

0.2

0.4

CH /

Nk

1/X = kT/B

Magnetic Heat Capacity

• In fact, this is a general result for the heat capacity in any two level system.

Isolated Paramagnetic Solid• Now consider a very similar problem to the one just

discussed. Now, constrain (fix) the total energy U of the isolated system.

• N total dipoles, n spin-up aligned with the applied B field:

• U is obviously a function of n.

• A given energy U(n) corresponds to a given number of n spin up atoms with statistical weight:-

)2()()( nNBBnBnNnU

!)!(

!)(

nnN

NnW

• So, the Entropy is given by:

• For large N (~1023) Stirling’s Approximation can be used:

• The Temperature of the spin system is given by:

!)!(

!ln)(ln)(

nnN

NknWknS

)ln()(lnln)( nNnNnnNNknS

n

nS

dU

dn

n

nS

nU

nS

U

S

T

)(1)(

)(

)(1

• Further manipulation gives:

• Also, we know that:

• So that:

• Now, solve for n, to find the density of spin up atoms:

n

nN

B

k

n

nS

BTln

2

)(

2

11

xeZN

n

1

1

dU

dn

n

nS

nU

nS

U

S

T

)(

)(

)(1

)2()( nNBnU

Negative Temperatures?• From the previous derivation,

• Manipulate this to obtain:

• Note that, if n < N/2 then more than half the dipoles are anti-parallel to the field & we also get the surprising result that

T becomes negative!

nN

n

B

k

n

nN

B

k

Tln

2ln

2

1

n

nN

B

k

n

nS

BTln

2

)(

2

11

What is a Negative Temperature?• First note that, as the temperature T , the populations

of spin-up & spin-down particles become equal!

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Pro

ba

bilit

y

X=B/kT

Spin aligned (U = -B) Spin antialigned (U = +B)

• We just saw that if n < N/2 (more spins down than up),

T becomes negative!This is called a population inversion.

• A negative temperature state must therefore be “hotter” than T , because it is a higher energy state of the system than T !

• For a negative temperature the entropy & statistical weight must be decreasing functions of E.

What is Negative Temperature?• For a negative temperature the entropy and statistical weight

must be decreasing functions of E.• This can happen if the system possess a state of finite

maximum energy – such as our paramagnet with U=NB.• No systems exist where this happens for all particular

aspects (I.e. vibrational energies, electronic energies and magnetic energies).

• However, if one such aspect or subsystem is effectively decoupled from the others, so they do not interact, that subsystem may be considered to reach internal equilibrium without being in equilibrium with the others.

• This is the case for magnetic systems where the relaxation times between atomic spins is much quicker than the relaxation between spins and the vibrational modes of the lattice.

Negative Temperature• In the paramagnet, the lowest possible energy is

U = -NB and the highest U = +NB. These are both unique microstates so S = 0.

• In between we can only reach states with positive energy with a negative temperature!!!

-5 -4 -3 -2 -1 0 1 2 3 4 5

-1.0

-0.5

0.0

0.5

1.0

En

erg

y /

NB

Temperature

System Energy

-20 -10 0 10 20

-1.0

-0.5

0.0

0.5

1.0

En

erg

y /

NB

1 / Temperature

System Energy