These are the notes that I’m taking for myself in order to learn some statistical mechanics.

The main reference is: *Anton Bovier, Lecture notes on Gibbs measures and phase transitions* (Bonn University)

In thermodynamics one has:

where are respectively the mechanical, chemical and thermal components of the energy.

is also denoted by ,

where is the presure and is the volume

where is the number of molecules, and is the average chemical potential per molecule

where is the temperature and is the volume

Thus we have, for a closed system:

We will view the total energy as a function of three **extensive** variables : . Then we have

are called **intensive** variables. The last three equations are called **equations of state**.

Instead of viewing as a function of three variables one can also view it as a function of another set of three variables, e.g. , etc. We will assume that is a convex function, so that change is possible. (Legendre transformation). Problem: it can’t be convex ?! Howevern can have a nondegenerate Hessian ?!

Intorduce a function (of three variables ) such that

So in fact

. This sunction is called the **Gibbs free energy**. Some other useful functions are:

Helmholtz** free energy**:

**Enthalpy**:

In some case is linear in : is a constant (then we can’t compute as a function of ). This situation is called a **first order phase transition**. (Example: condensation of vapor into water).

**1-dimensional model**:

particle, volume , pressure , each particle has moment

Total energy function (conserved quantity): is a constant,

Assume equi-distribution of all possible positions and momenta having a fixed , the level which has the highest probabality will be (assuming very large): . (This is the level that one will observe in reality).

The probability formula for is:

where

, .

Equilibrium state:

Everything works very nicely in this 1-dimensional model ! This model also indicates why the entropy is a log function, and what does it mean to be a micro-state.