The three principles of thermodynamics in a statistical context

With all the machinery of statistical physics, it becomes possible to prove, or further justify the three principles of classical thermodynamics

A remark: the conservation of information

zeroth principle and equilibrium

the principle zero of thermodynamics

First principle

\[U = \langle E \rangle = \sum_i p_i E_i\] \[\text{d} U = \sum_i E_i\text{d}p_i + \sum_i p_i \text{d}E_i\] \[\boxed{\delta Q = \sum_i E_i\text{d}p_i }\] \[\boxed{\delta W = \sum_i p_i \text{d}E_i}\]

Second principle

Third principle