For the grand canonical ensemble, obtain an expression for the ensemble average energy and ensemble average particle numbers.
In the grand canonical ensemble, a system is in contact with a reservoir that can exchange both energy and particles. The grand canonical partition function is defined as [ \mathcal{Z} = \sum_{N=0}^{\infty} \sum_{{i}} e^{-\beta (E_i - \mu N)}, ] where (E_i) is the energy of the microstate (i) with (N) particles, (\mu) is the chemical potential, (\beta = _________ _________ __________ _________ _________ ___.
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What is the Bose-Einstein condensation? Write an expression of number of particles in the excited state (Nex) in terms of Bose-Einstein condensation temperature (Tc), total number of bosons (N) in an assembly and temperature T. Show a plot of distribution of bosons as a function of temperature below and above the Tc.
The Hamiltonian for the non-relativistic, non-interacting, monoatomic ideal gas is given by
where m is the mass of the particle and U(qi) is the potential energy. Obtain an expression of the phase space volume of the energy shell and hence obtain an expression of number of microstates.
State and prove quantum Liouville equation.
Discuss the concept of thermodynamic fluctuations in the canonical ensemble. Define energy fluctuations and derive an expression for the mean square fluctuation of energy. Using a classical ideal gas, show that the relative fluctuation in energy varies as and hence, justify why thermodynamic calculations are valid for ordinary macroscopic systems.
Obtain expression for average energy for the canonical ensemble in terms of partition function Z.
The time evolution of the quantum state, is given by the Schrödinger wave equation:
What is meant by Cluster Integrals? Express B2 and B3 in terms of Cluster Integrals.