Category: Statistical Mechanics

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}

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