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Distribution function of molecules in presence of gravity is as function of height is derived. 

For an ideal gas Maxwell's distribution of velocities is obtained using canonical ensemble.

It is shown that the statistical entropy coincides with the thermodynamic expression for entropy.

In this lecture canonical ensemble and canonical partition function are introduced. This topic has a central place in equilibrium  statistical mechanics. This ensemble describes the microstates of a system in equilibrium with a heat bath at temperature \(T\). The probability of a microstate having energy \(E\) is proportional to \(\exp(-\beta E\), where \(\beta=kT\) and \(k\) is Boltzmann constant. 

$\newcommand{\pp}[2][]{\frac {\partial #1}{\partial #2}}$For a system in contact with heat bath, the energy is not well defined. The system being in a micro state with energy value \(E_k\) is given by Boltzmann distribution. We compute average energy and show that the average energy  it in terms of the  canonical partition function is given by \[ U = -k \pp[\ln Z]{\beta}.\] It is shown that for macroscopic systems the variance of energy is negligible compared to the energy, but not for microscopic systems. Hence \(\bar E\) can be identified with the thermodynamic internal energy.

Using the principle of maximum entropy, we derive conditions so that an isolated system consisting of  two systems may be in equilibrium.
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The postulate of equal a priori probability, the Boltzmann equation and the principle of increase of entropy and their role in statistical mechanics are briefly explained.