Effect of topology on the thermodynamic limit for a string gas

We discuss the thermodynamic limit for a gas of strings at high energy densities. This is defined by studying the statistical properties of the gas in a compact space and taking the size of the space to infinity keeping the energy density finite. We obtain a behavior that is different from earlier treatments where the gas is considered at the same energy density but living in a noncompact space. In particular we show that the gas is not dominated by a single energetic string above the Hagedorn energy density, but instead the number of energetic strings is $\frac{\ln R}{\sqrt{a^{\prime }}}$ where $R$ is the radius of the universe and ${\alpha }^{\prime }$ the slope parameter. The reason for the thermodynamic behavior being sensitive to topology is the existence of winding modes that can sense the large-scale structure of the space.