## Pathria Statistical Mechanics Homework Solutions

Statistical Mechanics - Homework Assignment 3

Alejandro G´omez Espinosa

∗

March 4, 2013

Pathria 3.41

A system of N spins at a negative temperature

(

E >

0)

is brought into contact with an ideal-gas thermometer consisting of N’ molecules. What will the nature of their state of mutual equilibrium be? Will their commom temperature be negative or positive, and in what manner will it be aﬀected by the ratio

N

/N

?

If the system of N spins has negative temperature its energy must be bounded. Then, if it is broughtinto contact with a sistem of N’ molecules, the equilibrium will be reach when both temperatures arethe same. But since the thermometer does not have bounded energy, the temperature in equilibriumshould be greater than zero.Then, by conservation of energy we can write:

−

Nε

tanh

εkT

+32

N

kT

=

E

T

(1)where we can see that the ratio

N

/N

will not be aﬀected by this result.

Pathria 4.8

Determine the grand partition function of a gaseous system of magnetic atoms (with

J

=

12

and

g

= 2

) that can have, in addition to the kinetic energy, a magnetic potential energy equal to

µ

B

H

or

−

µ

B

H

, depending on their orientation with respect to an applied magnetic ﬁeld H. Derive an expression for the magnetization of the system, and calculate how much heat will be given oﬀ by the system when the magnetic ﬁeld is reduced from H to zero ar constant volume and constant temperature.

In this case, the Hamiltonian of the system is given by:

H

i

=

p

2

i

2

m

i

±

µ

B

H

(2)Then, using the partition function for one particle and taking only the positive magnetic potentialenergy:

Q

+1

=

e

−

βE

d

3

p d

3

q h

3

=

e

−

β

p

22

m

+

µ

B

H

d

3

p d

3

q h

3

=

e

−

βµ

B

H

h

3

e

−

β

p

22

m

d

3

p d

3

q

=

Ve

−

βµ

B

H

h

3

∞−∞

e

−

β

p

22

m

d

3

p

=

Ve

−

βµ

B

H

h

3

2

mβ

√

π

3

=

2

mπβh

2

3

/

2

Ve

βµ

B

H

∗

gomez@physics.rutgers.edu

1

**Instructor: Roland E. Allen **

**Room M213 Mitchell Institute Building **(Fall, 2017)

**allen@tamu.edu, http://people.physics.tamu.edu/allen/StatMech.html** [phone removed by department as economy measure]

**Office hours: Monday, Tuesday, Wednesday ****3-4 p.**m.

**3-4 p.**

**The class meets each Tuesday and Thursday at 11:10 a.m. in MPHY 109. ****Each Tuesday at 4:00 p.m. we will have an optional homework presentation meeting, also in MPHY 109.**

**The class meets each Tuesday and Thursday at 11:10 a.m. in MPHY 109.**

**Each Tuesday at 4:00 p.m. we will have an optional homework presentation meeting, also in MPHY 109.**

## Textbook: *Statistical Mechanics*, 3rd Edition, by R. K. Pathria and P. D. Beale.

You may want to bring your copy of the textbook to class for easier note-taking.

## Evaluation:

## Homework 40%

2 “midterm” exams 40%

final exam (comprehensive) 20%

## Homework is due at the beginning of class each Tuesday.

Homework late by < 48 hours, 1/2 credit. Homework late by > 48 hours, no credit.

## The “midterm” exams and final exam will be in the evening at 7:30 - 10:30 p.m. in Room MPHY 213 on the designated dates: October 5, November 16, December 8.

**Some topics in astrophysics will be amplified compared to the textbook's treatments, with short and simple treatments of neutron stars and white dwarfs (including the Chandrasekhar limit), stellar pulsation, the Saha ionization equation, and reactions in cosmology. Topics not in **the textbook include (i) review of some techniques in thermodynamics (from Herbert B. Callen, *Thermodynamics and an Introduction to Thermostatistics*, 2nd Edition ), (ii) basics of the BCS theory of superconductivity (Linda E. Reichl, *A Modern Course in Statistical Physics*, 4th edition, pp. 224-231), and (iii) the Rayleigh-Bénard instability as an example of a nonequilibrium phase transition (Reichl, pp. 378-385).

## HOMEWORK [P means Pathria & Beale (the textbook), M means *Statistical Mechanics* by D. A. McQuarrie, and G means *States of Matter* by D.L. Goodstein -- each assignment should be regarded as tentative rather than final until two weeks before it is due.

## August 31: M1-58, 1-59

## September 5: P 1.9; M 1-28, 1-30, 1-29, 1-35

## September 12: P 1.7, 1.16; M 1-24, 1-61, 1-63

## September 19: P 2.1, M 1-51, Callen 3.4-11, Callen 3.4-12, Kubo 1.36

## September 26: P 3.6, P 3.7, P 3.15, P 3.16; M 2-17

## October 3: P 3.19, 3.20, 3.21 (a) classical only plus 3.22, 10.19, 3.25

## October 10: M 7-32, M 8-15, M 8-16, P 3.35 (a), P 3.43

## October 17: P 4.1, P 4.7, P 3.29 1st part, P 3.29 2nd part, M 10-4

## October 24: P 8.4 (a), P 8.6, P 8.8, P 8.13, M 10-19

## October 31: P 8.4 (b) and (c), P7.5, P 7.13 first paragraph, P 7.23, M 10-27

## November 7: P 6.1 first part, P 6.7, P 6.10, P 7.25, P 7.26

## November 14: P 9.8, P 11.1 (a), P 7.33, P 7.34, P 7.35

## November 21: P 12.1, P 11.8 (counts as 2 problems), G 5.12, G 6.3 (b) & 6.4 (counts as 1 problem)

## For extra credit (maximum credit - 2 homework sets): 10-minute talks on Tuesday, December 5, at 7:00 p.m., in our classroom MPHY 109.

## Some of the many possible very broad topics for talks: sample-607-topics.pdf .

**Phys607Exam1-2017.pdf** , Phys607Exam1-2017solution.pdf

**Phys607Exam2-2017.pdf** , Phys607Exam2-2017solution.pdf

**Phys607Final-2017.pdf** , Phys607Final-2017solution.pdf

**Exam1-2017-607.pdf** , Exam1-2017-607-solution.pdf

**Exam2-2017-607.pdf** , Exam2-2017-607-solution.pdf

**Final-Exam-2017-607.pdf** , Final-Exam-2017-607-solution.pdf

.

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