Large and small density approximations to the thermodynamic Bethe ansatz

We provide analytical solutions to the thermodynamic Bethe ansatz equations in the large and small density approximations. We extend results previousl y obtained for leading order behaviour of the scaling function of affine To da field theories related to simply laced Lie algebras to the non-simply la ced case. The comparison with semi-classical methods shows perfect agreemen t for the simply laced case. We derive the Y-systems for affine Toda field theories with real coupling constant and employ them to improve the large d ensity approximations. We test the quality of our analysis explicitly for t he sinh-aordon model and the (G(2)((1)), D-4((3)))-affine toda field theory . (C) 2000 Elsevier Science B.V. All rights reserved Science B.V. All right s reserved.