Hot String Soup

TL;DR

Above the Hagedorn energy density, the paper develops a Boltzmann-equation framework for long fundamental strings, showing that weak interactions distribute energy among many long strings rather than collapsing into a single string. The equilibrium solution yields $n(\ell)=e^{-\ell/\bar{L}}/\ell$ and a single-string density of states $\omega(\varepsilon)=e^{\beta_H \varepsilon}/\varepsilon$, with the microcanonical ensemble predicting $\Omega(E)\sim e^{\beta_H E}$ and an average of $\sim \log E$ long strings. Open-string dynamics suppresss long-string formation in the thermodynamic limit, while closed strings can form a stable, interacting long-string ensemble that can split and join. The work reconciles Boltzmann-equation results with free-state calculations on compact target spaces, highlights the importance of finite-volume and gravitational effects (Jeans instability), and provides a basis for non-equilibrium string thermodynamics with potential cosmological applications.

Abstract

Above the Hagedorn energy density closed fundamental strings form a long string phase. The dynamics of weakly interacting long strings is described by a simple Boltzmann equation which can be solved explicitly for equilibrium distributions. The average total number of long strings grows logarithmically with total energy in the microcanonical ensemble. This is consistent with calculations of the free single string density of states provided the thermodynamic limit is carefully defined. If the theory contains open strings the long string phase is suppressed.