What is the universal definition of the order parameter that is valid irrespective of the nature of the phase transition?

Question

Plausible definition Consider a phase transition from phase 1 to phase 2. The order parameter is zero in one of the phases 1 or 2 and nonzero in the other.

For example, in normal (phase 1) to superfluid (phase 2) transition, the order parameter is zero in the normal phase and nonzero in the disordered phase. So in this case, the above definition works good.

However, in the case of gas (phase 1) to liquid (phase 2) transition, the order parameter is taken to be $\mathcal{O}=\rho_{liq}-\rho_{gas}$. But $\mathcal{O}$ is nonzero in both the phases 1 and 2, and only vanishes above the critical temperature $T_c$. So, in this case, the above definition doesn't hold good.

Does it mean that the definition

Consider a phase transition from phase 1 to phase 2. The order parameter is zero in one of the phases 1 or 2 and nonzero in the other.

is wrong?

Is there a universal definition of order parameter such that it hold's good in both the cases?

Answer 1

The order parameter is discontinuous for first order phase transition at the transition point. However, it need not be zero in any of the phases between which the first order transition takes place. For example, the liquid-to-gas transition or vice-versa below $T_c$, the order parameter is $\rho_{liq}-\rho_{gas}$, is non-zero in both liquid and gas but changes discontinuously. This also holds good for first order magnetic transition from up-aligned Ising ferromagnet to the down-aligned phase.

However, the same order parameter $\rho_{liq}-\rho_{gas}$ is zero in the phase above $T_c$ (the disordered phase) and nonzero below it for the second order phase transition at $T_c$. This also works for paramagnetic to ferromagnetic transition at $T_c$. This resolves the problem.