## Overview: Statistical mechanics theory

In this chapter you will learn the theoretical underpinnings of the statistical thermodynamics that you started to study in the previous chapter more rigorously. You will learn about the principle of equal apriori probabilities that underpins this theory and about the relationship between the entropy in statistical mechanics and the information in information theory. You will then learn how we can construct a generalized partition function by performing a constrained optimization of the information. You will then see how we can use this generalized ensemble to construct thermodynamic ensembles that can be used to describe systems that are under a variety of different constraints on intensive and extensive quantities. To consolidate all that you have learnt you will learn how we can create models that describe interacting and non-interacting gases and adsorption of molecules on surfaces. Lastly, you will learn about the statistical mechanics of first and second order phase transitions.

### Aims

• You should be able to state the principle of equal apriori probabilities.
• You should be able to state Khinchines axioms for information theory and you should be able to use these axioms to derive an expression for the information content of a probability distribution.
• You should be able to derive an expression for the probability of being in a particular microstate in the generalized ensemble by performing a constrained optimization of the information using Lagrange's method of undetermined multipliers.
• You should be able to write out expressions for being in a microstate in the canonical, microcanonical, isothermal isobaric and grand canonical ensembles.
• You should be able to derive the equation of state for an ideal gas by first calculating the canonical partition function. You should also be able to explain the assumptions that are made about the particles of which the gas is composed in this model.
• You should be able to explain the equipartition principle and when this partition can be applied.
• You should be able to derive the Langmuir adsorption isotherm.
• You should be able to discuss how the microscopic coordinates of the particles behave when a system undergoes a phase transition.

### Contact Details

School of Mathematics and Physics,
Queen's University Belfast,
Belfast,
BT7 1NN

Email: g.tribello@qub.ac.uk
Website: mywebsite