Statistical Mechanics

A short series of posts on the subject of statistical mechanics.

Mea culpa: I’m not particularly proud of these posts. They are unoriginal and would serve as a very poor introduction to statistical mechanics for (say) an undergraduate. I wrote them primarily for myself, so I could make better sense of my (probably atypical) 2nd year stat mech course. The problem is my failure to address where the Gibbs entropy (the maximisation of which forms the basis of all ensuing theory) really comes from.

Despite my shame, I have left these posts up because I invested quite a lot of time in their writing.

0. Introduction – a preamble outlining the need for statistical mechanics.

1. Defining the Gibbs Entropy – a terse outline of the origin of the Gibbs entropy.

2. Lagrange Multipliers – an incidental sketch proof of the method of Lagrange multipliers for a single constraint.

3. Isolated Systems – the state probability distribution for the so-called ‘microcanonical ensemble’.

4. Knowledge – a brief post relating macroscopic observables to constraints.

5. Closed Systems – the state probability distribution for the so-called ‘canonical ensemble’.

6. Partition Function – a post defining one of the most important functions in statistical mechanics. Its utility will be shown in later posts.

7. First Law of Thermodynamics – an expression for the conservation of energy for closed systems.

8. Additivity of the Gibbs Entropy – a short post showing how entropy combines for two systems in contact.

9. Thermodynamic Beta – what does it mean? A few clues are provided in this post.

10. Interpreting the Gibbs Entropy – a long post attempting to decipher entropy. The first half is largely mathematical, and the second, wordy half focuses on an interpretable example.

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11. Mini Partition Function – we derive an approximation of the partition function of a population of non-interacting, identical particles (something nearly as exciting as it sounds)

12. Non-Interacting Two-Level Particles – a simple but instructive example.

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