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

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

State Ising model and obtain the partition function (ln Z) for 1D in the absence of an external magnetic field up to second order perturbation. Explain the physical significance of each term in the Hamiltonian and discuss the role of the coupling constant J.

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

i) Uniform 

ii) Rayleigh distribution 

How many photons are present in 3 1cm of radiation at 727^oC? What is their average energy?

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

Show that for any distribution where N is large, the distribution tends to a Gaussian or a normal distribution.