LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034  M.Sc. DEGREE EXAMINATION – PHYSICS

AC 11

FIRST SEMESTER – NOV 2006

         PH 1806 – STATISTICAL MECHANICS

(Also equivalent to PH  1801)

 

 

Date & Time : 26-10-2006/1.00-4.00           Dept. No.                                                       Max. : 100 Marks

 

 

 

PART A ( 20 MARKS )

 

ANSWER ALL QUESTIONS.                                        10 X 2 = 20

 

1. State the ergodic hypothesis. Is it true?
2. What is meant by stationary ensemble?
3. When is the classical limit of the quantum description of systems valid?
4. State the condition for mechanical equilibrium between two parts of a composite system.
5. State two features of the Gibb’s canonical ensemble.
6. What is the significance of the temperature T₀ for an ideal Bose gas?
7. Does the chemical potential of an ideal Fermi gas depend on temperature?
8. What is the implication of Einstein’s result for the energy fluctuations of black body radiation?
9. What is a stationary Markoff process?
10.  Write down the Boltzman transport equation.

 

PART B ( 30 MARKS )

 

ANSWER ANY FOUR QUESTIONS.                     4 X 7.5 =30

 

 

 

1. State and explain the basic postulates of statistical mechanics.

 

1. Obtain the distribution for an ideal Fermi gas.

 

 

1. Apply the Bose- Einstein statistics to photons and obtain the Planck law of black body radiation.

 

1. Discuss the temperature dependence of the energy, specific heat and entropy of an ideal Bose gas.

 

 

1. Calculate the concentration fluctuation for a grand canonical ensemble. Show that for an ideal classical gas it increases as the volume of the gas decreases.

 

 

 

 

 

 

 

 

PART C ( 50 MARKS )

 

 

ANSWER ANY FOUR QUESTIONS.                                    4 X 12.5 = 50

 

 

 

1. (a) Prove Liouville theorem. Use it to arrive at the principle of conservation of density in phase space.

(b) Explain the principle of conservation of extension in phase space.

 

17.Calculate the entropy of an ideal Boltzman gas using the micro canonical ensemble. Explain the corrections to be made to obtain the Sackur-Tetrode equation.

 

18.Calculate the pressure exerted by a Fermi-Dirac gas of relativistic electrons in the ground state. Use the result to explain the existence of the Chandrasekhar limit on the mass of a white dwarf.

 

19.Discuss Brownian motion in 1-d and obtain an expression for the particle concentration as a function of (x,t). Explain how Einstein estimated the particle diffusion constant.

 

20. Derive the Boltzmann transport equation. Use it to find the distribution function in the absence of collisions.

 

 

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034  M.Sc. DEGREE EXAMINATION – PHYSICS

AC 15

FIRST SEMESTER – NOV 2006

         PH 1810 – STATISTICAL MECHANICS

 

 

Date & Time : 26-10-2006/1.00-4.00           Dept. No.                                                       Max. : 100 Marks

 

 

 

PART A (20 MARKS)

 

 

ANSWER ALL QUESTIONS                                      10 X 2 = 20

 

 

1. State the ergodic hypothesis.
2. State the principle of conservation of extension in phase space.
3. When is the classical limit of the quantum description of a system valid?
4. Sketch the Maxwell velocity distribution.
5. Why is the super fluid transition in Helium known as the lambda transition?
6. What is the significance of the fermi temperature?
7. What is the pressure exerted by a Fermi gas at absolute zero?
8. How is the super fluidity of Helium-3 explained?
9. Give Einstein’s relation for the particle diffusion constant.
10. Define spectral density for a randomly fluctuating quantity.

 

PART B (30 MARKS)

 

 

ANSWER ANY FOUR QUESTIONS                         4 X 7.5 = 30

 

 

1. Discuss the quantum picture of a micro canonical ensemble.

 

1. Obtain the distribution for an ideal Fermi gas.

 

 

1. Apply the Bose  Einstein statistics to photons and obtain the Planck law for black body radiation.

 

1. Find the temperature dependence of the chemical potential for an ideal FD gas.

 

 

1. Discuss the random walk problem in 1-d and apply the results to a system of N particles each having a magnetic moment m.

 

 

 

 

 

 

 

 

PART C (50 MARKS)

 

 

ANSWER ANY FOUR QUESTIONS.                                    4 X 12.5 = 50

 

 

1. Calculate the entropy of an ideal Boltzmann gas using the micro canonical ensemble. Explain the corrections to be made to obtain the Sackur-Tetrode equation.

 

1. (a) Discuss the features of the Gibb’s canonical ensemble.

(b) Discuss the rotational partition function for a system of diatomic molecules.

 

1. Discuss the thermodynamic properties of an ideal Bose-Einstein gas.

 

1. Calculate the pressure exerted by a FD gas of relativistic electrons in the ground state. Use the result to explain t5he existence of Chandrasekhar limit on the mass of a white dwarf.

 

 

1. (a) Show that the fractional fluctuation in concentration is smaller than the MB case for FD statistics and larger for BE statistics.

(b) Obtain Einstein’s result for the energy fluctuations of black body radiation. What is the implication of the result?

 

 

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