# Primon Gas

Today I’m going to be talking about an interesting little toy model in statistical mechanics – the Primon Gas.

Consider a physical system with a discrete energy spectrum

$E = \{ \ln 2, \ln 3, \ln 5, ...\} = \{ \ln p : \text{p is prime} \}$

Each energy in the spectrum corresponds to a particle with that energy. If we second quantize this system, we obtain a creation operator $\alpha_{p}$ for each of these particles. Using these operators, we can act on a vacuum state (zero energy state), denoted $| 1 \rangle$, to obtain new states. We get the following ‘tower’ of states with corresponding energies:

$\begin{array}{rcl} \textbf{State} & \to & \textbf{Energy} \\ \alpha_2 | 1 \rangle & \to & \ln 2\\ \alpha_3 | 1 \rangle & \to & \ln 3 \\ \alpha_2 \alpha_2 | 1 \rangle & \to & \ln 4 \\ \alpha_5 | 1 \rangle & \to & \ln 5 \\ \vdots \end{array}$