Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 344 (Theoretical Physics of Soft Condensed Matter)

Use the Fourier transform conventionThe zero mode vanishes because the compositional order parameter has zero spatial average. Parseval identity and the Fourier transform of a derivative turn the quadratic part of the dimensionless free energy intoHere ${\sum }_{\mathbf{q}}^{+}$ is the positive-wavevector sum for a real field. Each independent complex amplitude has density proportional to $e^{-G(q)|{\varphi }_{\mathbf{q}}{|}^2}$, so its elementary Gaussian integral gives the static structure factorwhenever $G(q)>0$.

The stationary points of the Brazovskii model kernel obeyBecause $\kappa <0<\gamma$, the nonzero minimum is the nonzero-wavevector soft-mode sphereAt this wavevector,The first Gaussian field theory divergence therefore occurs at

For the one-mode smectic phase ansatz $\varphi (\mathbf{r})=A\cos (q_{0}z)$, take $A\ge 0$ without loss of generality. Spatial averaging givesandUsing $G(q_0)=a-a_c$, the free-energy density isFor $a\ge a_c$, its minimum is $A=0$. For $a<a_c$, the nonzero stationary point isandThus the modulation amplitude grows linearly and continuously from zero below $a_c$. In the language of an order-parameter critical exponent, this nonanalytic $|\varphi {|}^3$ theory has the mean-field value $\beta =1$.

This is the Gibbs--Bogoliubov--Feynman inequality. In units with $k_{B}T=1$, $H$ is the exact dimensionless Hamiltonian, here the full interacting free-energy functional, and $H_0$ is a freely chosen trial Hamiltonian. Its partition function and free energy areThe notationmeans an expectation in the trial Gibbs ensemble. The bound is therefore

Choose the translationally invariant Gaussian variational approximationThe superscript $+$ means that one member of each pair $\{\mathbf{q},-\mathbf{q}\}$ is included; this is necessary because a real field obeys ${\varphi }_{-\mathbf{q}}={\varphi }_{\mathbf{q}}^{\ast }$. The constrained zero mode is omitted.

For each independent complex mode,Consequently,Substitution into the Gibbs--Bogoliubov--Feynman inequality giveswhich is the required upper bound.

Translational invariance makes the pointwise variance independent of $\mathbf{r}$. Applying Parseval identity to the stated Fourier convention givesIn the thermodynamic limit, the reciprocal-lattice sum becomesAt each point $\varphi (\mathbf{r})$ is a centered Gaussian random variable, so the supplied absolute third moment gives

SetThe factor two in the last expression restores the omitted negative wavevectors. For one independent mode $J_k=J(k)$,Differentiating the bound from parts (d) and (e) givesIts stationary point therefore obeys the self-consistency equationThe interaction has generated a positive, wavevector-independent mass shift.

Write the optimized kernel asNear the soft-mode sphere, the supplied asymptotic result saysThus the self-consistency equationcannot reach $\bar{a}=a_c$ for any $g>0$: the fluctuation correction diverges first. The isotropic static structure factor therefore remains finite, and the isotropic state never undergoes the continuous Gaussian instability predicted in part (a).

The modulated minimum from part (b), however, has negative free energy for sufficiently low $a$ and amplitude $A_{\ast }=\pi (a_c-a)/(8g)$. For small positive $g$, its free energy must cross that of the still locally stable isotropic variational state. At the crossing the isotropic inverse susceptibility is positive and the smectic amplitude is nonzero. The order parameter consequently jumps, which is the Brazovskii fluctuation-induced first-order transition.

The thermodynamic order parameter is the conserved scalar composition difference $\varphi$ of the binary fluid mixture. The momentum density $\rho \mathbf{v}$ is an additional conserved hydrodynamic mode; it is not a symmetry-breaking order parameter, but it must be retained because momentum relaxes only through spatial transport.

Composition conservation gives a continuity equation,The current $\mathbf{J}=-M\nabla \mu$ is the leading isotropic, dissipative constitutive law: it drives material down gradients of the chemical potential. For the Landau-Ginzburg theorythe functional derivative isThese statements give the advective Cahn--Hilliard equation.

Constant mass density and incompressible flow require $\nabla \cdot \mathbf{v}=0$. Momentum conservation gives the Navier-Stokes equation: material acceleration equals the sum of the Newtonian viscous force $\eta {\nabla }^2\mathbf{v}$, the pressure force $-\nabla P$, and the Korteweg force $-\varphi \nabla \mu$. The pressure is the Lagrange multiplier enforcing incompressibility. Together these equations are Model H dynamics; isothermality removes the need for a separate energy equation.

Assume late-stage bicontinuous phase separation has one characteristic domain size $L(t)$ and statistically self-similar morphology. The dynamical scaling of binary-fluid coarsening then estimatesThe interface has surface tension $\sigma$. Its curvature pressure is of order $\sigma /L$, so its coarse-grained gradient, equivalently the transverse part of $-\varphi \nabla \mu$, scales as $\sigma /L^2$. Diffusive composition transport is neglected because hydrodynamic advection controls the late stage, and $\rho ,\eta ,\sigma$ are treated as constants.

The pressure cannot simply be discarded: in an incompressible flow it cancels the longitudinal part of the other forces and contains both capillary and dynamic contributions. Applying the divergence-free projection to the momentum equation eliminates $\nabla P$ while retaining a solenoidal force with the same scaling. Geometry, projection, signs, and correlations are absorbed into dimensionless order-one coefficients. The result is

SetThen $\dot{L}=(L_0/t_0)f^{\prime }$ and $\ddot{L}=(L_0/t_0^2)f^{\prime \prime }$. Every term in the scaling equation has the common dimensions and magnitude ${\rho }^2{\sigma }^3/{\eta }^4$. Cancelling that factor yields the dimensionless equationor equivalently

For a power law $f(u)\sim u^n$,Independence of $\rho$ requires $n=1$, givingThis is viscous hydrodynamic coarsening. The viscous and capillary terms both scale as $u^{-2}$, whereas the convective inertial term scales as $u^{-1}$. It is therefore subleading for $u\ll 1$.

Independence of $\eta$ requires $n=2/3$, givingThis is inertial hydrodynamic coarsening. Both inertial terms and the capillary term scale as $u^{-4/3}$, while the viscous term scales as $u^{-5/3}$ and is subleading for $u\gg 1$.

The crossover occurs when $u$ is order one. Hencewith a dimensionless order-one constant $C_X$ determined by the coefficients and morphology.

For an arrested domain size $\Lambda$, make the prescribed replacementWritinggives the schematic algebraic equationor

For $w\ll 1$, viscous-capillary balance giveswhich is independent of $\rho$. For $w\gg 1$, inertial-capillary balance giveswhich is independent of $\eta$. These are the two limits of the stirring-arrested binary-fluid domain size.

Inertia dominates viscosity whenequivalently when the imposed root-mean-square velocity gradient satisfies ${\tau }^{-1}\ll t_0^{-1}$. In the inertial-capillary regime,The corresponding Reynolds number isThe geometry is therefore not fixed as the stirring rate changes. Slower stirring permits much larger domains, and the growth of ${\Lambda }^2$ more than offsets the reduction of $U/\Lambda$. This is why the inertial regime occurs at low imposed velocity gradient in this self-adjusting coarsening problem.

For a specified trajectory, the nonconserved order-parameter dynamics equation determines the noise realizationThe forward Onsager--Machlup path probability for Model A dynamics is therefore

Assume the order-parameter field is even under time-reversal symmetry and its free-energy functional is time-reversal invariant. The reversed path isIts time derivative changes sign, soFor additive Gaussian white noise, the trajectory-to-noise Jacobian is the same in the two directions. We assume it and all path-independent normalization factors are absorbed into equal constants $N_F=N_B$. A time-reversal-odd order parameter would require the corresponding parity transformation as well.

Subtracting the two quadratic actions givesThe functional chain rule identifies the last integral as $F_2-F_1$, so

Microscopic time-reversal invariance implies detailed balance. The equilibrium probability density of a configuration with free energy $F$ obeysThereforeComparison for arbitrary endpoint free energies yields the Model A fluctuation-dissipation relation

After the sudden parameter change, the functional derivative isEach Cartesian component of each Fourier mode consequently obeysThis is an Ornstein-Uhlenbeck process. The integrating-factor method givesBecause both coefficients are positive, every mode has positive decay rate.

Initial equilibrium at $a_I$ givesThe initial field and later noise are independent, so their cross terms vanish. DefineWith the supplied Gaussian white noise covariance, the noise contribution isUsing the decay rate found in part (c), it follows thatThis is the covariance interpolation in a Gaussian Model A quench: each mode forgets its initial equilibrium with relaxation time $1/r(q)$ and approaches the final equilibrium variance. Larger-$q$ modes relax faster.

LetEvolution from the earlier time to the later time multiplies the earlier field by the deterministic decay factor, plus fresh noise independent of that field. The unequal-time correlation function is thereforeEquivalently,When both observation times are many relaxation times after the quench, the second term vanishes and the stationary Ornstein-Uhlenbeck process covariance remains: