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 

0

at which the chemical potential of the gas is zero ( 

z

= 1

). Express 

0

in terms of the Fermi temperature 

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

Let us start with Equation (8.1.4):

N V 

=

3

3

/

2

(

z

) =(2

πmkT 

)

3

/

2

h

3

gf 

3

/

2

where

=

0

as defined in the problem. Then, solving for

0

:

0

=

2

/

3

h

2

2

πmk

(

gV f 

3

/

2

(

z

))

2

/

3

(1)Then, the Fermi temperature is defined by

=

ε

k

where the Fermi energy

ε

is given by theequation (8.1.24):

ε

=

3

4

πgV 

2

/

3

h

2

2

m

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

0

=

2

/

3

h

2

2

πmk

(

gV f 

3

/

2

(

z

))

2

/

3

=

4

π

3

3

/

2

(

z

)

2

/

3

3

4

πgV 

2

/

3

h

2

2

mπk

=

4

π

3

3

/

2

(

z

)

2

/

3

πk

=

43

√ 

πf 

3

/

2

(

z

)

2

/

3

(3)Now, let us calculate the factor

3

/

2

(

z

). Since

µ

= 0, therefore

z

=

e

µβ

= 1, thus

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

=

1

12

 j

ζ 

(

 j

+ 1)

gomez@physics.rutgers.edu

1

PHY831 - Graduate Statistical Mechanics: Fall 2012

Lectures MWF 11:30-12:20, BPS1308
  • Professor Phillip Duxbury: Office: BPS 4260, Email: duxbury at pa.msu.edu
  • Helproom - TBA
  • Office hours - email me for an appointment

    Active lecture notes and problem sets.

    Complete Lecture Notes and Problems for Part 4
    Complete 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.
  • Monday September 10: Homework 1 is due and Quiz 1 (20 minutes) in class
    Homework 1 - do problems 1-9, hand in problems 8,9. Quiz 1 covers quiz problems 1-8 and homework problems 1-9.
  • Monday September 17: Homework 2 is due and Quiz 2 (20 minutes) in class
    Homework 2 - do problems 10-16, hand in problems 15,16. Quiz 2 covers quiz problems 9-18 and homework problems 10-16.
  • Monday September 24: Midterm I (50 minutes) in class
    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.
  • Friday October 5: Homework 3 is due and Quiz 3 (20 minutes) in class
    Homework 3 - do problems 1-9, hand in problems 6, 8. Quiz 3 covers quiz problems 1-9 and homework problems 1-9.
  • Monday October 15: Homework 4 is due and Quiz 4 (20 minutes) in class
    Homework 4: hand in solutions to assigned problems 10,14
    Quiz 4 covers quiz problems 9-18 and assigned problems 10-16
  • Monday October 22: Midterm II (50 minutes) in class
    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.
  • Monday November 5: Homework 5 is due and Midterm 2 - take 2 in class
    Homework 5: hand in solutions to assigned problems 5, 6
  • Monday November 12: Homework 6 is due, Quiz 5 and Quiz 6 (20 minutes each) in class
    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.
  • Monday November 19: Midterm 3 (50 minutes) in class
    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
  • Monday December 3: Homework 7 is due and Quiz 7 (20 minutes) in class
  • Friday December 7: Midterm 4 (50 minutes) in class


    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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