Statistical mechanics in just one equation

 This semester, I am giving four lectures in a third-year undergraduate course on statistical mechanics. Last year, I gave a guest lecture on the Ising model.

The first thing I want to emphasise is that the whole course is built around just one equation.

exp(−F(T, V)/kT) = ∑_s exp(−E_s/kT) ≡ Z(T, V)

This connects macroscopic thermodynamic properties (contained in the Helmholtz free energy F(T,V)) to microscopic properties (the energies E_s of all possible states s of the system).

[Note that from equilibrium thermodynamics, partial derivatives of F(T,V) give the entropy (and specific heat capacity) and the pressure (equation of state)].

I think that we are so used to this equation that we may miss just how amazing and profound it is.

First, the equation is incredibly simple.

Second, it is universal. It applies to any system in thermodynamic equilibrium regardless of its chemical or physical composition.

Third, in the context of the theory of emergent phenomena, it is exceptional because it provides a robust, tested way to connect the microscopic to the macroscopic. Biology, neuroscience, economics, sociology, and computer science have nothing like it.

Fourth, although the above three points are impressive, the basis of its validity remains an outstanding problem (a mystery?). One can "derive" it and "justify" it by drawing on assumptions such as the fundamental postulate ("in an isolated system all accessible microstates are equally probable"), the principle of maximum entropy, and the validity of equilibrium thermodynamics. But why are they true?

Although the equation is simple, implementing it, particularly for systems of interacting particles, is challenging. This challenge can be broken into five steps. One can get stuck on any one of the steps. People build whole careers on them.

1. For a system of interest, propose microscopic states of each particle and a Hamiltonian for the whole system.

2. Enumerate all possible microscopic states of the system.

3. Evaluate the energy of each of these microstates.

4. Perform the sum over all states in the partition function Z.

5. Take the thermodynamic limit where the system size becomes infinite.