We study the free energy of the (1 + 1)-dimensional O(N) nonlinear sigma -m
odels for even N using the TBA equations proposed recently. We give explici
t formulae for the constant solution of the TBA equations (Y-system) and ca
lculate the first two virial coefficients. The free energy is also compared
to the leading large N result. (C) 2001 Elsevier Science B.V. All rights r
eserved.