GLYPH<Minus> 6 GLYPH<Minus> 5 GLYPH<Minus> 4 GLYPH<Minus> 3 GLYPH<Minus> 2 GLYPH<Minus> 1 0.02 0.04 0.06 0.08 0.10 0.12 0.14 c for repulsive fermions PSfrag replacements c µ/T FIG. 20: The scaling function c for repulsive fermions as a function of µ/T . The free energy density then takes the simple form: F = − 1 β ∑ a ∫ ( d k ) [ − s$_{a}$ log ( 1 − s$_{a}$e − βε a ) − 1 − y − 1 a 2( e βε a − s$_{a}$ ) ] (106) For the two-component fermion defined by the action (3), spin up particles scatter with spin down, and G$_{↑↓}$ = G$_{↓↑}$ . Thus, if the chemical potentials are equal, µ$_{↑}$ = µ$_{↓}$ = µ , then ε$_{↑}$ = ε$_{↓}$ , and the thermodynamics is just two copies of the single component fermion described above. In particular, the density and c are doubled, but the energy per particle scaling functions are the same. X. CONCLUSIONS We have shown that the S-matrix based approach to quantum gases developed in [20] leads to a new treatment of the scale-invariant unitary limit, where all of the thermodynamic scaling functions can be computed as a function of µ/T . Though our 40