LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M. Sc. DEGREE EXAMINATION – PHYSICS

FIRST SEMESTER – NOVEMBER 2003

PH 1801 / PH 721 – STATISTICAL MECHANICS

06.11.2003                                                                                              Max.   : 100 Marks

1.00 – 4.00

 

PART – A

Answer ALL the questions.                                                                         (10 x 2 = 20)

 

1. Distinguish between micro states and macro states.

 

2. What is meant by stationary ensemble?

 

3. Distinguish between canonical and grand canonical ensembles.

 

4. How does the vibrational contribution to the specific heat of a system vary with temperature?

 

5. What is the significance of the temperature T_o for an ideal Bose-Einstein gas?

 

6. Sketch the Fermi-Disc distribution function for a gas in 3-d at T = 0 and at T > 0.

 

7. What are white dwarfs?

 

8. What is the implication of Einstein’s result for the energy fluctuations of black body radiation?

 

9. Give the relations which represent the Wiener-Khintchine theorem.

 

10. Write down the Boltzmann transport equation.

 

 

PART – B

 

Answer any FOUR questions.                                                                                (4 x 7.5 = 30)

 

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

 

12. Obtain the Sackur-Tetnode equation by considering and ideal gas in canonical ensemble.

 

13. Apply the Bose-Einstein statistics to photons and obtain Planck’s law for black body radiation. Hence obtain the Stefan-Boltzmann law.

 

14. Show that the specific heat of an ideal Fermi-Dirac gas is directly proportional to temperature when T << T_F.

 

15. Calculate the energy fluctuation for a canonical ensemble. Show that if the fluctuations are very small, it is practically a micro canonical ensemble.

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PART – C

 

Answer any FOUR questions.                                                                              (4 x 12.5 = 50)

 

16. a) Prove Liouiville theorem.

 

7. b) Explain Gibbs paradox and discuss how it is resolved. (5 + 7.5)

 

17. a) Show that Boltzmann counting appears as a natural consequence of the symmetry of wave function in quantum theory.                          (5)

 

7. b) Discuss the features of Gibbs canonical ensemble. Derive an expression for the probability distribution of the canonical ensemble. (7.5)

 

18. a) Discuss the thermodynamic properties of an ideal Bose-Einstein gas. (7.5)

 

1. b) How does Landau explain the super fluidity of He⁴ using the spectrum of phonons and rotons?              (5)

 

19. a) Show that the fractional fluctuation in concentration is smaller than the MB case for FD statistics and larger for BE statistics. ( 7.5)

 

1. b) State and explain Nyquist theorem.              (5)

 

20. Obtain the Boltzmann transport equation. Using it determine the distribution function in the presence of collisions.

 

 

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             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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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

FG 28

M.Sc. DEGREE EXAMINATION – PHYSICS

FIRST SEMESTER – APRIL 2008

    PH 1810 / 1801 – STATISTICAL MECHANICS

 

 

 

Date : 05/05/2008            Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

PART A ( 20 MARKS )

ANSWER ALL QUESTIONS. Each question carries 2 marks.

 

1. State the ergodic hypothesis. Is it true ?
2. Distinguish between the micro-canonical ensemble and the canonical ensemble.

3. State the postulate of equal-a-priori probability.

1. Sketch the Maxwell velocity distribution
2. How does the vibrational contribution to the specific heat vary with temperature ?
3. What are quasi-particles ? Give an example.
4. What is the pressure exerted by an ideal Fermi gas at absolute zero ?
5. What is the importance of the Chandrasekhar limit ?
6. What is the implication of Einstein’s result for the energy fluctuations of blackbody radiation ?
7. State Nyquist theorem.

 

    PART B ( 30 MARKS )

ANSWER ANY FOUR QUESTIONS. Each question carries 7.5 marks.

 

1. State and explain the basic postulates of statistical mechanics.
2.  Obtain the distribution for an ideal Maxwell –Boltzmann gas.
3.  Explain Bose-Einstein condensation. Discuss the super-fluidity of Helium by considering it as a form of Bose-Einstein condensation.
4. Derive the Richardson-Dushman equation, which describes thermionic emission.
5.  Obtain the relations, which state the Wiener-Khintchine theorem.

 

PART C ( 50 MARKS )

ANSWER ANY FOUR QUESTIONS. EACH QUESTION CARRIES 12.5 MARKS.

 

1. (a)  Explain Gibb’s paradox. How is it resolved ?

(b) Prove Liouiville theorem.

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

 

1.  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 for a white dwarf.
2.  (a) Calculate the concentration fluctuations for a grand canonical ensemble. Show that for an ideal classical gas it increases as the volume of the gas increases.

(b)   Prove the Nyquist theorem.

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – PHYSICS

FIRST SEMESTER – NOVEMBER 2010

    PH 1815 / 1810  – STATISTICAL MECHANICS

 

 

 

Date : 09-11-10                 Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

Part – A

Answer all questions                                                                                                   ( 10 x 2 = 20 )

1. What is the form of second law of thermodynamics when the number of particles of the system under observation is not a constant?
2. What is meant by correct Boltzmann counting?
3. Evaluate the integral
4. What is grand canonical potential? Express grand canonical partition function in terms of the potential.
5. Why does liquid ³He show super fluidity even though ³He molecules are Fermions.
6. The pressure exerted by a Boson gas below the critical temperature is independent of its volume. Substantiate this statement.
7. Why does the electronic heat capacity dominate the atomic heat capacity at very low temperatures?
8. Is nuclear matter degenerate or not? Justify your answer.
9. Define mean square deviation.
10. The ensemble of a large system approximates a microcanonical ensemble. Substantiate this statement.

 

Part – B

Answer any four questions                                                                                      ( 4 x 7.5 = 30 )

11. a) What is a Slater determinant? How is Pauli’s exclusion principle incorporated into the Fermion wave function? (5)
12. b) In a one dimensional box of length 2a, a particle with constant velocity is mirror reflected at the walls. Draw its phase trajectory. (2.5)
13. Obtain the grand canonical distribution function.
14. Derive Stefan Boltzmann law for black body radiation.
15. Derive an expression for the magnetic susceptibility of a free electron gas.
16. Obtain an expression for the concentration fluctuation in grand canonical ensemble.

 

Part – C

Answer any four questions                                                                                      ( 4 x 12.5 = 50 )

16. a) State and prove Liouville’s theorem.    (10)
17. b) Explain the principle of conservation of extension in phase. (2.5)
18. Obtain the expression for the entropy of an ideal gas by the method of canonical ensemble.
19. What is Bose-Einstein condensation? With necessary theory and relevant diagram show how the BE distribution function varies as temperature decreases below the transition temperature.
20. Explain the theory for the specific heat capacity of liquid helium below transition temperature.
21. Applying the theory of one dimensional random walk, show that a system of Brownian particles concentrated at the origin x=0 at time t=o, spread out with time.

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – PHYSICS

FIRST SEMESTER – APRIL 2012

PH 1815 – STATISTICAL MECHANICS

 

 

Date : 02-05-2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

PART – A

Answer ALL questions:                                                                                           (10×2=20)

 

1. What is the degree of freedom of a system of N diatomic molecules
2. What is meant by correct Boltzmann counting?
3. What is the statistical weight associated with the distribution , for a grand canonical ensemble.
4. Differentiate between density of states g(є) and degeneracy g_i.
5. Why does ³He show super-fluidity even though it is a Fermion?
6. What would be the pressure exerted by a Boson gas on the walls of the container at absolute zero? Justify your answer.
7. Define the term Fermi energy.
8. What is meant by thermionic emission? Define work function of a metal.
9. Why is statistical thermodynamics unsuitable for a small system at low temperatures?
10. Define the correlation function for a randomly fluctuating quantity.

 

PART – B

Answer any FOUR questions:                                                                                             (4×7.5 = 30)

 

11. Explain Gibb’s paradox. How is it resolved?
12. Obtain the partition function of a system with rotational, vibrational and electronic degrees of freedom.
13. Derive Planck’s radiation law. Show that the partition function for an oscillator defined by
14. Derive an expression for the magnetic susceptibility of a free electron gas.
15. Obtain Einstein’s result for the energy fluctuations of black body radiation. What is the implication of the result?

PART – C

Answer any FOUR questions:                                                                                             (4×12.5 = 50)

 

16. State and prove Liouville’s theorem. Express the equation of motion of a phase point in Poisson’s bracket notation.
17. a) Obtain Grand canonical distribution function. (6.5)
18. b) Consider an ideal gas in a grand canonical ensemble. Show that its fugacity is directly proportional to concentration. (6)
19. Explain the phenomenon of BE condensation. Why do only Bosons and no other particles exhibit it? Show how the distribution of Bosons varies with temperature.
20. Show that the specific heat capacity of an ideal Fermi gas is directly proportional to temperature when the temperature is very small compared to its Fermi temperature

 

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

 

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