# Stat Mech Pathria Homework

Statistical Mechanics - Homework Assignment 5

Alejandro G´omez Espinosa

∗

April 28, 2013

Pathria 8.2

For a Fermi-Dirac gas, we may deﬁne a temperature

T

0

at which the chemical potential of the gas is zero (

z

= 1

). Express

T

0

in terms of the Fermi temperature

T

F

of the gas. (Hint: Use equation (E.16).)

Let us start with Equation (8.1.4):

N V

=

gλ

3

f

3

/

2

(

z

) =(2

πmkT

)

3

/

2

h

3

gf

3

/

2

where

T

=

T

0

as deﬁned in the problem. Then, solving for

T

0

:

T

0

=

N

2

/

3

h

2

2

πmk

(

gV f

3

/

2

(

z

))

2

/

3

(1)Then, the Fermi temperature is deﬁned by

T

F

=

ε

F

k

where the Fermi energy

ε

F

is given by theequation (8.1.24):

ε

F

=

3

N

4

πgV

2

/

3

h

2

2

m

(2)Hence, comparing (2) and (1):

T

0

=

N

2

/

3

h

2

2

πmk

(

gV f

3

/

2

(

z

))

2

/

3

=

4

π

3

f

3

/

2

(

z

)

2

/

3

3

N

4

πgV

2

/

3

h

2

2

mπk

=

4

π

3

f

3

/

2

(

z

)

2

/

3

F

πk

=

43

√

πf

3

/

2

(

z

)

2

/

3

T

F

(3)Now, let us calculate the factor

f

3

/

2

(

z

). Since

µ

= 0, therefore

z

=

e

µβ

= 1, thus

f

3

/

2

(

z

= 1) =1Γ

32

∞

0

x

1

/

2

e

x

+ 1

dx

(4)That is an integral easy to calculate using equation (E.16):1Γ(

j

+ 1)

∞

0

η

j

e

η

+ 1

dη

=

1

−

12

j

ζ

(

j

+ 1)

∗

gomez@physics.rutgers.edu

1

## PHY831 - Graduate Statistical Mechanics: Fall 2012

**Lectures**MWF 11:30-12:20, BPS1308

### Active lecture notes and problem sets.

Complete Lecture Notes and Problems for Part 4Complete solutions to problems for Part 4

Helproom: Thursday Dec. 6, 7-8 pm BPS 1308

Midterm 4: Friday Dec. 7, usual time and place

Helproom: Wednesday Dec. 12, 11:30am - 12:30pm BPS 1308. All of the homeworks, quizzes and midterms you handed in will be available.

**Final and Subject Exam:**Friday Dec. 14, 2-5pm, BPS 1400

*Final covers Parts 1-3 and BCS theory at finite temperature in Part 4*

### Course schedule and outline

**Part 1: (LL, PB) Foundations: (10 lectures)**

History, Heat, Engines, Kinetic theory and Entropy. Computational methods, molecular dynamics, ensembles, ergodicity. Foundations of Equilibrium Statistical Mechanics. Ensembles, Boltzmann factors, Quantum systems. Computational methods, Monte Carlo and detailed balance. Free energies and thermodynamics. Fun with thermodynamic relations. Fluctuations and response functions.

*Homework 1 - do problems 1-9,*

**hand in problems 8,9. Quiz 1 covers quiz problems 1-8 and homework problems 1-9.***Homework 2 - do problems 10-16,*

**hand in problems 15,16. Quiz 2 covers quiz problems 9-18 and homework problems 10-16.**Complete Lecture Notes and Problems for Part I

Complete Problems and Solutions for Part I

Midterm I Exam and Solutions

**Part 2: (H, PB, LL) Key solvable systems: (10 lectures)**

Non-interacting spin systems. Ideal Classical gas. Classical harmonic oscillators. Ideal Fermi gas, electron gas, white dwarf stars. Ideal Bose gas, photons, phonons, bose condensation, The early universe. Electrons and phonons in metals. Solvable Ising systems. Magnetic properties of the electron gas.

*Homework 3 - do problems 1-9,*

**hand in problems 6, 8. Quiz 3 covers quiz problems 1-9 and homework problems 1-9.****Homework 4: hand in solutions to assigned problems 10,14**

**Quiz 4 covers quiz problems 9-18 and assigned problems 10-16**

Complete Lecture Notes and Problems for Part 2

Complete Problems and Solutions for Part 2

Midterm 2 Exam and Solutions

**Part 3: (H, PB) Interacting systems, phase transitions and critical phenomena (11 lectures)**

Interacting spin systems, Ising model. Interacting classical gas, cluster expansion, van der Waals gas, virial Expansion. BCS model of superconductivity. Scaling theory, universality, Landau theory, Ginzburg-Landau model.

**Homework 5: hand in solutions to assigned problems 5, 6**

**Homework 6: hand in solutions to assigned problems 8, 9**

**Quiz 5 and Quiz 6 cover Lectures 1-7, Quiz problems 1-17 and Assigned problems 1-11**

**You will not be asked to prove the linked cluster theorem in these quizzes.**

**You will not be asked to prove the linked cluster theorem. You will not be asked to reduce the BCS mean field Hamiltonian to diagonal form using the Bogolubov-Valatin transformation.**

Complete Lecture Notes and Problems for Part 3

Solutions to problems for Part 3

Midterm 3 Exam and Solutions

**Part 4: (H, PB) Scaling and complex systems (7 lectures)**

Landau theory of phase transitions and critical phenomena, scaling. Equilibrium and non-equilibrium dynamics

**Final Exam:**Friday December 14, 2pm - 5pm (BPS 1400)

### Course assessment

Weekly or biweekly homeworks (10%). Hand in a copy of your work, not the original. Late homeworks will not be accepted without a written explanation.Quizzes (random, almost weekly) (20%). Your worst quiz will be dropped.

Midterms (40%). Final (30%).

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