statmechlecture 1.49.03 AM

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

Info iconThis preview shows pages 5–15. Sign up to view the full content.

View Full Document Right Arrow Icon
• We want to find, p i . Given <E i >.
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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:
Background image of page 6
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.
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Background image of page 8
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.
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Example 1: Ideal Gas Law
Background image of page 10
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 12
Example 2: Force-Displacement relation for a chain
Background image of page 13

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 14
Background image of page 15
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page5 / 15

We want to find p i Given E i Means Gibbs Entropy Recipe is...

This preview shows document pages 5 - 15. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online