statmechlecture 1.49.03 AM

We want to find p i given e i means gibbs entropy

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We want to find, p i . Given <E i >.
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Means: Gibbs Entropy Recipe is to maximize Gibbs entropy Subject to constraints on the system Hence find p i . Once we know p i . We can find any other quantity B as:
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Boltzmann Distribution •Consider a system at temperature, T in thermal equilibrium with the heat bath. •The thermodynamic energy, of the system is constant. •We maximize Gibbs entropy subject to this constraint. We use the method of lagrange multipliers. Writing a modified entropy Normalization refers to the fact that, Maximizing w.r.t p i and using the normalization constraint we will get.
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Partition Function The denominator of is the most important quantity. It is called partition function. It is denoted by Z. Substituting the p i obtained in Gibb’s entropy formula, we get, Differentiating S w.r.t <E> and using above eqn, we get
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Properties of partition function The free energy of the system is hence: Thus derivative of ln Z w.r.t to the lagrange multiplier gives corresponding average quantity.
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Example 1: Ideal Gas Law
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Example 2: Force-Displacement relation for a chain
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