Thermodynamic Properties of the Piecewise Uniform String

TL;DR

This paper calculates the thermodynamic free energy of a quantum excitation gas on a piecewise uniform bosonic string with different tensions and lengths, revealing how the Hagedorn temperature scales with string parameters.

Contribution

It provides an explicit integral expression for the free energy of a piecewise uniform string under specific tension and length ratios, extending understanding of string thermodynamics.

Findings

Hagedorn temperature scales with the square root of the length ratio s.

Explicit integral form of free energy for the piecewise string is derived.

Analysis under the tension ratio approaching zero.

Abstract

The thermodynamic free energy F is calculated for a gas whose particles are the quantum excitations of a piecewise uniform bosonic string. The string consists of two parts of length L_I and L_II, endowed with different tensions and mass densities, adjusted in such a way that the velocity of sound always equals the velocity of light. The explicit calculation is done under the restrictive condition that the tension ratio x = T_I/T_II approaches zero. Also, the length ratio s = L_II/L_I is assumed to be an integer. The expression for F is given on an integral form, in which s is present as a parameter. For large values of s, the Hagedorn temperature becomes proportional to the square root of s.