Assume a binary mixture, with the component $A$ in concentration $c$ and the component $B$ in concentration $1-c$. The total energy $E(c)$ is thus given by a function of one variable $c$ (if we do not take into account the kinetic energy...). Now let's assume that energy has two minima at $c_0$ and $c_1$. The minimization of energy should lead to a phase segreggation.

But : In order for two phases $1$ and $2$ to be at equilibrium, we need two conditions : $\mu_1=\mu_2$ and $\pi_1=\pi_2$. Two equations for 1 variable !! The system is overconstrained and there cannot be any phase segregation. How come the minimization of the energy and basic themodynamics not compatible ?


Actually my question was mostly wrong. There are indeed 2 variables : $\phi_1$ of the first phase and $\phi_2$ of the second phase. So we need 2 equations.

However, if the second phase $\phi_2$ is imposed (if we want the second phase to be at some given), then there will be overconstrained, and the potential will have to be "chosen" ad hoc, so that the constraints do not contradict each other.

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