Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 344 (Theoretical Physics of Soft Condensed Matter)

For a binary fluid mixture, the scalar compositional order parameter $\varphi (\mathbf{r})$ may be taken as the local concentration difference between its two molecular species. Its spatial integral is fixed by their separately fixed total amounts:Locally, $\varphi$ can change only through transport and therefore obeys a continuity equation. By contrast, $\mathbf{p}$ is the local polar order parameter measuring the mean tail-to-head orientation of the surfactant molecules. Individual molecules can rotate in place, so the integral of $\mathbf{p}$ need not be conserved.

The composition changes most rapidly across an interface, so $\nabla \varphi$ points along its normal. The termtherefore couples the surfactant's head-to-tail orientation to the interface normal. Minimization aligns $\mathbf{p}$ antiparallel to $\nabla \varphi$ when $\lambda >0$ and parallel when $\lambda <0$. The sign is fixed by which fluid component is called positive $\varphi$ and by which molecular end defines positive $\mathbf{p}$, together with the preferential affinity of the head and tail for the two components.

The terms involving $\mathbf{p}$ can be written by completing the square:Translation of the integration variable in the Gaussian functional integral over $\mathbf{p}$ contributes only a $\varphi$-independent determinant. The effective free energy is thereforewhereThis is the surfactant renormalization of the square-gradient coefficient.

Surfactants accumulate at an interface and orient their two chemically distinct ends toward their preferred fluid components. The relaxation found in part c lowers the coefficient of the gradient energy, and hence generally lowers the surface tension. If the reduction makes $\tilde{\kappa }<0$, the positive fourth-gradient term proportional to $\gamma$ can stabilize structure at a nonzero wavevector, producing the modulated correlations characteristic of a microemulsion.

Use ${\partial }_x\mapsto iq$ under the Fourier transform and the reality conditions $\varphi (-q)=\varphi (q{)}^{\ast }$ and $p(-q)=p(q{)}^{\ast }$. With $\Psi =(\varphi ,p)$, the quadratic free energy iswhereThe opposite imaginary off-diagonal entries make $G(q)$ a Hermitian matrix. Reversing the Fourier-sign convention reverses both of those signs without changing any correlator.

The covariance of a centered multivariate Gaussian distribution is the inverse of its quadratic kernel. Writing $A(q)=a+\kappa q^2+\gamma q^4$, one findsConsequently the composition static structure factor is

The other diagonal entry of the same inverse givesor, more revealingly,The first term is the uncoupled local orientational fluctuation. The second shows that every nonuniform composition fluctuation induces a correlated surfactant-polarization fluctuation. It vanishes at $q=0$, as the coupling contains a gradient, and is enhanced at wavevectors where the composition structure factor is large.

A real symmetric traceless $2$ by $2$ matrix has eigenvalues $\lambda$ and $-\lambda$ and can be writtenfor a unit nematic director $\mathbf{n}\sim -\mathbf{n}$. Since $Q_{ij}{Q}_{ji}=2{\lambda }^2$, the uniform Landau free energy isFor $a<0<b$, its nonzero minima satisfyand henceThe signs $\pm {\lambda }_0$ amount to exchanging the two orthogonal eigenvectors, so the chosen convention takes ${\lambda }_0\ge 0$.

The bulk potential gives fluctuations in $\lambda$ a nonzero restoring force, whereas slowly rotating the nematic director costs only gradients. At lengths much larger than the amplitude correlation length, it is therefore the leading gradient expansion approximation to set $\lambda ={\lambda }_0$ while retaining $\mathbf{n}(\mathbf{r})$.

DifferentiatinggivesSubstitution into the elastic term yields

Let $\mathbf{n}=(\cos \theta ,\sin \theta )=(c,s)$ with $\theta =\theta (x)$. ThenThe squared norm is $({\theta }^{\prime }{)}^2$ becauseThus the one-elastic-constant nematic free energy becomeswith

The anchoring conditions are $\theta (0)=0$ modulo $\pi$ and $\theta (L)=\pi /2$ modulo $\pi$, since the nematic director identifies angles differing by $\pi$. Their difference can therefore be $m\pi /2$ for any odd integer $m$. The Euler-Lagrange equation ofis ${\theta }^{\prime \prime }=0$, so every stationary solution has the formIts free energy per unit length in the $y$ direction isThe smallest possible $m^2$ is one, giving exactly the two degenerate global minima $m=1$ and $m=-1$.

Traverse the large rectangle counterclockwise, taking its lower edge at $y\to -\infty$ and upper edge at $y\to +\infty$. Along the lower edge the director angle changes by $-\pi /2$; along the upper edge, traversed from $x=L$ to $x=0$, it changes by another $-\pi /2$. The anchored director is constant along the two vertical edges. The net continuous angle change is thereforeThe topological charge of a two-dimensional nematic disclination enclosed by a circuit is $q=\Delta \theta /(2\pi )$, soA nonsingular director field on the enclosed disk would have zero winding. At least one nematic disclination must therefore lie inside.

The least costly configuration contains one charge-$-1/2$ nematic disclination, placed near $(L/2,0)$ by the reflection symmetries of the boundary data. Away from its small core, the director interpolates as smoothly as possible between the wall anchoring and the two far-field textures. Locally around the defect one may sketchThe elastic energy of an isolated defect scales as $q^2\log (R/a_{\mathrm{core}})$. Splitting the required total $-1/2$ charge into additional allowed half-charge defects is impossible without also adding compensating defects, which raises both the logarithmic elastic energy and the positive core energy. The single centered $-1/2$ defect is therefore the lowest-energy topology, up to smooth distortions and symmetry-related core placement.

For general odd $m_{+}$ and $m_{-}$, the same rectangular circuit givesEvery elementary two-dimensional nematic defect has $|q|=1/2$, so the minimum number is

In three dimensions the director takes values in the real projective plane, whose fundamental group is $\mathbb{Z}/2\mathbb{Z}$. The signs of half-charge line defects are no longer distinct topological classes, and two such lines can annihilate by escape into the third dimension. Hence only the parity of $N_{2D}$ remains:When one is required, it is a disclination line extending through the unbounded $z$ direction.

Put $E(t)=-\delta A/\delta \mathbf{x}(t)$. For a specified forward trajectory, the equation of motion determineswhereas its time reverse requiresAssume the action is invariant under time-reversal symmetry, the position is even and velocity is odd under time reversal, and the trajectory-to-noise Jacobian is identical in the two directions. The Onsager--Machlup path probability is thenwith $\zeta \dot{\mathbf{x}}$ replaced by $-\zeta \dot{\mathbf{x}}$ for ${\mathbb{P}}_B$. Sincetheir ratio is

The functional derivative of the action isFor the Hamiltonian functionand a time-independent Lagrangian,Therefore

Detailed balance says that each equilibrium transition is balanced by its time reverse:With the Boltzmann distribution $P_{\mathrm{eq}}\propto e^{-\beta H}$ and $\Delta H=H_2-H_1$, this requiresParts a and b instead give $\exp [-(2\zeta /{\sigma }^2)\Delta H]$. Equality for every pair of endpoints yields the fluctuation-dissipation relation for a Langevin particle

With an external force $\mathbf{F}$, the required forward and backward noise histories becomeRepeating the difference of squares and using $2\zeta /{\sigma }^2=\beta$ givesThe second term is $\beta$ times the work performed by the external force.

The first law of thermodynamics, with $\Delta Q$ defined as heat lost by the particle to the bath, isThus $W-\Delta H=\Delta Q$, and part d becomes the local detailed balance identityThe logarithmic path-probability ratio is the bath's entropy production $\Delta S_{\mathrm{bath}}=\Delta Q/T$ in units of $k_B$.

If $\nabla \times \mathbf{F}=0$ on a simply connected region containing the bounded trajectory, then the Poincare lemma gives a scalar potential $U$ with $\mathbf{F}=-\nabla U$. The work iswhich remains bounded when $\mathbf{x}$ and $\mathbf{F}$ remain bounded. Since $H$ is also bounded, $\Delta Q=W-\Delta H$ cannot grow linearly with the observation time.

Thus sustained linear heat dissipation in this setting requires a nonconservative force, and, under the stated simply connectedness assumption,Such a force can perform nonzero work on repeated bounded cycles. On a multiply connected domain, a curl-free force can have nonzero circulation, so the topological assumption is essential.

For a specified field trajectory, the required normalized noise isand time reversal changes only $\dot{\mathbf{p}}$ to $-\dot{\mathbf{p}}$. Assuming equal additive-noise Jacobians, the difference of the two Onsager--Machlup actions givesThe functional chain rule identifies the first term as $-\beta \Delta F$, soThe forcing performs generalized work $W_Y=\int \mathbf{Y}\cdot \dot{\mathbf{p}}$, and $W_Y-\Delta F$ is the heat dissipated into the bath. The formula is therefore the field-theory form of local detailed balance and quantifies nonequilibrium entropy production.