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.

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

Using the result   show that the expression of density matrix of a 

free particle in box of volume V in the canonical ensemble in the coordinate representation is given by

Obtain Fermi-Dirac distribution function. Draw it for  What is its physical significance? 

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

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

i) Uniform 

ii) Rayleigh distribution