Statistical Physics A Final exam two
25 February 2021
Please solve the problems and send scans with your own solutions via e-mail to:
byczuk@fuw.edu.pl before 13:30 pm. Write in the subject: SPA EXAM TWO
Each problem is worth 10 points. Explanations and comments in writing, alongside mathema- tical formulation, are necessary and will count to final points. All problems must be solved by yourself and without any help from any other persons. In case of similar solutions both or more persons will have to be interrogated. It is allowed to use your own notes, lecture notes, and books. If you have questions or comments during the test time, please send them via e-mail to all of the teachers: maciej.lisicki@fuw.edu.pl, marta.waclawczyk@fuw.edu.pl and byczuk@fuw.edu.pl. They will reply with answers, which will be send to all students via USOS e-mail. Therefore you are recommended to check your e-mails regularly during the test time.
Good luck!
Problem 1. Ultrarelativistic gas An ultrarelativistic gas (for which the particle dispersion relation reads E = c|~p|, where ~p is the momentum of a particle and c is a velocity constant) is in contact with an energy and particles reservoir characterized by constant temperature T and the chemical potential µ. Find the grand canonical partition function and determine the chemical potential of the gas as a function of temperature and pressure, i.e. µ(T, p). Treat the particles classically.
Problem 2. Compressibility of photons The compressibility is defined as κ = −V (∂p/∂V )T and describes how the system responds when its volume is changed. Find the compressibility of (a) classical ideal particles and (b) quantum gas of photons. Explain both results on physical grounds. (Warning: physical discussion is important in final evaluation of the problem.)
Problem 3. Three-spin molecules A single triangle molecule is composed of three spins 1/2
A magnetic moment 2µsi with each i-th spin, where µ is a positive constant (Bohr magneton) and si = ±1/2 with i = 1, 2, 3. The molecule is placed in an external magnetic field B. In the Ising model, used to represent this system, the Hamiltonian is
H = J s1s2+ J s2s3− 2µB (s1+ s2+ s3) ,
where J is the interaction constant. A gas of such molecules is in equilibrium at the tempera- ture T .
(a) Write down all possible microstates and their corresponding energies for a molecule.
(b) Find the canonical partition function Z1(T, B) for a molecule.
(c) Consider a special case of the absence of the external field (B = 0). What are probabilities of occurence of microstates for a molecule when T → 0? How do they change for kBT J ? (d) Consider a molecule in a weak, but nonvanishing magnetic field (0 < B J/µ). What are probabilities for different microstates in the limit T → 0?
(e) Consider a molecule in a strong field B J/µ. What are probabilities for different micro- states in the limit T → 0?
(f) Consider now an ideal gas of N such molecules. Find the mean energy and the total magnetic moment of the gas in the following cases:
(1) vanishing magnetic field B = 0,
(2) vanishing magnetic field B = 0 in the limit of kT J , (3) weak, non-zero magnetic field B J/µ,
(4) strong magnetic field B J/µ.
Problem 4. Maxwell distribution Assuming a Maxwell probability distribution of the velocity of a single gas particle (of mass m) at temperature T , find a normalized probability distribution ρT(T ) of the kinetic energy T (treated as a random variable) of this particle. Using this distribution calculate the mean value and the most probable value of the kinetic energy and the mean square fluctuation of the kinetic energy.
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