Obtain an expression of probability of finding the system in the microstate k corresponding to energy E_{k i}n canonical ensemble. Also, write the expression of canonical partition function.

Calculate relative fluctuations in the energy of the system about the mean value in canonical ensemble.

What is Gibb’s paradox? Derive Sackur-Tetrode equation of entropy for a classical, ideal gas.

Using the Mayer f-function, show that the second virial coefficient B₂ for  gas

   is the

diameter of the sphere. Here  

Consider a single classical 1-D non-relativistic harmonic oscillator. Obtain x(t) and p(t) as a function of time t. Assume that initially at t = 0, the oscillator is at the maximum displacement  .Plot its phase space trajectory for fixed total energy E

Consider a gas in lattice-gas model, in which gas contained in volume V is assumed to be divided into fixed small cubic cells of volume V0 such that each of the (cell) can contain at most one atom. The system behaves as a set of distinguishable and independent cells in contact with a reservoir at temperature T and chemical potential .Obtain the expressions of canonical partition function and average number of particles

Calculate the mean, and the variance of the following probability density functions:

i) Uniform 

ii) Rayleigh distribution