Statistical Physics MSQ - IIT JAM MCQ

As the levels are non-degenerate, there is only one state for each energy.
Let the number of particles occupying the 3 energy states be N₁, N_2, and N₃ respectively
where 
The particles are identifiable, the number of ways of choosing the particles is

The energy of system is

Now, the most probable distribution is the one in which W is a maximum, subject to constraint given by equation (1)
Thus if 
and  
If N₂ = 3, N₃ = 0
and N₁ = 5 – (3 + 0) = 2
No other distribution are possible
For  

So must probable distribution is  
The correct answers are:  
 

For a given kind, all particles are exactly the same and are thus indistinguishable.
For each degree of freedom, there is one position coordinate and one canonical momentum,
∴ for n degrees, 2n will be the dimension.
Photons follow Bose Einstein statistics.
The correct answers are: All particles of given kind are treated as mutually indistinguishable., The phase space for n degrees of freedom will have 2n dimensions and it unit cell volume in nⁿ.

At very low temperature/high energy densities, the concentration of particles with similar momentum (low) would violate the Pauli’s exclusion principle.
The correct answers are: Very low temperature, High density

Bosons are particles that carry force, there is an attractive force between bosons (except for photons carrier of electromagnetic force, they can be repulsive also.)
Fermions are particles that make up the matter, they due to Pauli’s exclusion principle exert repulsive force on each other.
It is due to these respective attractions and repulsions that at given temperature.
Bose gas has lower pressure as compared to both Fermi gas and classical gas and Fermi gas has higher pressure.
The correct answers are: A fermion gas will exert greatest pressure, A boson gas will exert the least pressure.

A grand canonical ensemble is such that the system can exchange energy and particles with a reservoir so that various possible states of the system can differ in both then total energy and total number of particles.
The correct answers are: A can exchange energy with B, A can exchange particles with B

Correct statements: b) Particles following F-D statistics are indistinguishable

• This is true. Fermions (particles following Fermi-Dirac statistics) are fundamentally indistinguishable particles d) Particles following B-E statistics are indistinguishable
• This is also true. Bosons (particles following Bose-Einstein statistics) are also indistinguishable

Incorrect statements: a) Particles following F-D statistics are distinguishable

• This is false. Fermions are indistinguishable c) Particles following Maxwell Boltzmann statistics are indistinguishable
• This is false. Maxwell-Boltzmann statistics applies to distinguishable particles

Key points:

• Both quantum statistics (F-D and B-E) deal with indistinguishable particles
• Classical M-B statistics deals with distinguishable particles
• The indistinguishability of particles is a fundamental quantum mechanical property

Therefore, statements b) and d) are true.

In a canonical ensemble, system can exchange only energy with environment.
Number of particles remains constant, also volume of the system remains constant. Therefore, temperature and density of system are constant.
The correct answers are: temperature is constant, density is constant

Liouville’s theorem is all about the conservation of density of points ρ is phase space.
i.e.  
⇒ ρ is conserved.
Also we know that in microcanonical ensemble, the system is isolated in the sense that it cannot exchange energy or particles with its environment.
Hence Liouville’s theorem holds only for microcanical ensemble.
The correct answer is: micro canonical ensemble

The ensemble average of any physical quantity is equal to its time average and holds for all the ensembles whether micro-canonical, canonical or grand canonical ensemble.
The correct answers are: a microcanonical ensemble, canonical ensemble, grand canonical ensemble

Electronic contribution of specific heat is completely independent of the dimension of the system and is always linear with the temperature.
The correct answers are: 3-dimensional electron gas, 2-dimensional electron gas, 1-dimensional electron gas