The size-scaling of this effect is semi-quantitatively investigated in Fig. . Note that E Dark -E LP ∝ √ N g c while the maximum entropy of the dark states is approximately k B log N , for large N . From Fig. it is evident that, for small N , the entropic contribution could dominate the free-energy ordering of polaritons, while at large N the energy gap between the lower polariton and the dark states will dominate. This can be also verified analytically by simply considering the ratio lim