Classical Limit of Bethe-AnsatzThermodynamics for the Sine-Gordon System

We use the quantum Bethe-Ansatz method to compute the free energy of the classical sine-Gordon system. Previous attempts to do this have failed because the number of coupled integral equations to be solved to find the free-energy diverges in the classical limit. We present a transformation, extending a method of Maki, which reduces this divergent set to only two coupled equations, for the densities of solitons and anharmonic phonons. These equations can be solved iteratively in the temperature $t$ and the soliton density $e^{\frac{-1\mathrm{}}{t}}$ to give a double series for the free energy. This series coincides to high order with classical transfer-matrix results.