Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 344 / 1

For a binary fluid mixture, the compositional order parameter $\varphi (\mathbf{r})$ may be taken as the local concentration difference between the two species. In a closed systemis fixed by the total amount of each species. This is the sense in which $\varphi$ is conserved; locally it changes through a current and obeys a continuity equation.

For a symmetric mixture, the Landau-Ginzburg theory free energy isA term $h{\int }_V\varphi ,d\mathbf{r}$ is constant on the allowed configurations and therefore changes neither equilibrium probabilities nor the dynamics. Equivalently, it merely shifts the chemical potential by a spatial constant, whose gradient vanishes.

The coupling $\frac{c}{2}\int \varphi p^2$ is linear in $\varphi$, but its coefficient $p^2(\mathbf{r})$ is generally position dependent and dynamical. It therefore cannot be reduced to a constant times the conserved integral of $\varphi$; it changes both the local chemical potential of the mixture and the tendency of the polar order parameter to order.

Interchanging the labels of the two species sends $\varphi \mapsto -\varphi$ while leaving the even terms in $\varphi$ unchanged. It sends $c\mapsto -c$, so one may choose $c>0$ without loss of generality.

The isotropic disordered system is invariant under spatial inversion, which sends the polar order parameter $p\mapsto -p$ while leaving the scalar composition $\varphi$ unchanged. A term $\varphi p$ is odd under this symmetry and is forbidden; in more than one dimension it also fails to be a rotational scalar. Finally, if no higher even powers are retained, $b<0$ or $B<0$ makes the free energy unbounded below as the corresponding field grows. Thermodynamic stability therefore requires $b,B>0$.

At fixed uniform $\varphi$, the nonconserved variable $p$ can relax freely, so mean-field approximation minimizeswith respect to $p$. The stationary solutions obeyThus $p=0$ when $A+c\varphi \ge 0$, while $p^2=-(A+c\varphi )/B$ when $A+c\varphi <0$. Substitution givesWriting this in the requested form yieldsbecause $A+c\varphi =c(\varphi -{\varphi }_o)$. The value ${\varphi }_o$ is the composition at which the coefficient of $p^2$ changes sign, so it is the mean-field threshold for polar ordering.

With ${\varphi }_o=0$,For $C<a$, both branches are locally convex at the origin. Atthe negative-side curvature vanishes; for $C>a$ the interval near $0^{-}$ has $f^{\prime \prime }=a-C+3b{\varphi }^2<0$, so a homogeneous composition there is unstable and the equilibrium free energy is its convex envelope. The sketch therefore has an ordinary upward quartic on the positive side and, beyond $C_c$, a negative-curvature shoulder and a minimum on the negative side.

The two binodal compositions ${\varphi }_1<0<{\varphi }_2$ are the contact points of the common-tangent construction. Equivalently, they solveThe first equality is equality of chemical potential; the second is equality of pressure. Heretogether with the intercept equation determines the two densities. As $C\downarrow a$, both contact points approach zero continuously with scale $|{\varphi }_{1,2}|\propto \sqrt{C-a}$, so this mean-field onset of phase separation is continuous.

When $b={\kappa }_1=0$, stationarity in $\varphi$ givesSubstituting this value, or completing the square in the Gaussian integral over $\varphi$, givesWhen $A>0$ and the quartic term is negligible, each Fourier mode is Gaussian with the Ornstein--Zernike correlation functionup to the chosen Fourier normalization. Its correlation length is ${\xi }_p=\sqrt{\kappa /A}$.

The eliminated field obeys $\varphi =-c{p}^2/(2a)$, so its connected correlation function isFor the zero-mean Gaussian field of part (i), Wick theorem givesIt is therefore nonnegative and has the square of the Ornstein--Zernike spatial form. In particular, if $C_p(r)$ has exponential factor $e^{-r/{\xi }_p}$, then $C_{\varphi }(r)$ has $e^{-2r/{\xi }_p}$ and correlation length ${\xi }_p/2$, with the algebraic prefactor also squared.