Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 344 (Theoretical Physics of Soft Condensed Matter)

Let $X$ be a coarse-grained configuration, with restricted partition function $Z_X$ and free energy $F(X)=-k_{B}T\log Z_X$. Its Boltzmann distribution is $\pi (X)\propto e^{-\beta F(X)}$. Microscopic reversibility pairs every equilibrium microscopic history with a history read backwards, including reversal of all time-odd variables. Summing this identity over the microscopic histories representing a given coarse-grained path preserves it; a Markov assumption is not needed for this argument.

Write ${\mathbb{P}}_F$ for the probability density of a specified history conditional on its starting configuration $X_1$, and ${\mathbb{P}}_B$ for its physically reversed history conditional on starting at $\Theta X_2$, where $\Theta$ reverses time-odd variables. Equilibrium time-reversal symmetry givesSince $F(\Theta X)=F(X)$, the desired conditional path relation isFor a reversible Markov chain, this also follows by multiplying detailed balance ratios $P_{ij}/P_{ji}={\pi }_j/{\pi }_i$ along the history: the intermediate equilibrium weights telescope.

The conditioning is essential. If the starting equilibrium weights are included in the path probabilities, the forward and reversed probabilities are equal, rather than differing by the displayed exponential. For example, a two-state continuous-time Markov chain with $F_1=0$, $F_2=\Delta >0$, rates $k_{12}=e^{-\beta \Delta }$ and $k_{21}=1$ has equal equilibrium-weighted probabilities for the two reversed single-jump histories. Their conditional rate ratio is instead $e^{-\beta \Delta }$. Normalized bridges conditioned on both endpoints also have a different normalization from the path weights used above. The PDF prints ${\mathbb{F}}_B$ in the denominator while its explanatory text defines ${\mathbb{P}}_B$; the latter is the intended notation. For the scalar order parameter used subsequently, $\Theta \varphi =\varphi$.

Here $\Gamma >0$ is a kinetic coefficient, and the additive Gaussian white noise must be centered. A rigorous starting point is a finite spatial discretization or finite set of modes of the coarse-grained field. For each field coordinate, the Fokker-Planck probability current has the continuum notationThe equilibrium Boltzmann distribution ${\mathcal{P}}_{\mathrm{eq}}\propto e^{-\beta F}$ has $\delta {\mathcal{P}}_{\mathrm{eq}}/\delta \varphi =-\beta (\delta F/\delta \varphi ){\mathcal{P}}_{\mathrm{eq}}$. Detailed balance for this time-even field requires zero configuration-space current. ThereforeThis is the Model A fluctuation-dissipation relation. Conversely, that noise strength makes the equilibrium current vanish and gives reversible relaxational dynamics when the equilibrium measure is normalizable.

Alternatively, the Onsager--Machlup path probability for Model A dynamics compares the squared required noise histories $\dot{\varphi }+\Gamma \delta F/\delta \varphi$ and $-\dot{\varphi }+\Gamma \delta F/\delta \varphi$. Their difference gives $\log ({\mathbb{P}}_F/{\mathbb{P}}_B)=-2\Gamma \Delta F/{\sigma }^2$, using the functional chain rule. The midpoint convention and time-reversal invariance of a path Jacobian justify cancellation of the common Jacobian. Comparing with the conditional path ratio gives the same result. The spatial white-noise continuum is a coarse-grained idealization: retain a microscopic cutoff when pointwise fluctuations or ultraviolet divergences matter.

With periodic or sufficiently decaying boundary conditions, integration by parts givesAssume $\kappa \ge 0$ for a stable quadratic free energy; the diffusive smoothing discussed below requires $\kappa >0$. In a periodic box of volume $V$, use the unitary convention ${\varphi }_{\mathbf{q}}=V^{-1/2}{\int }_V\varphi (\mathbf{r})e^{-i\mathbf{q}\cdot \mathbf{r}}d\mathbf{r}$. A real field has ${\varphi }_{-\mathbf{q}}={\varphi }_{\mathbf{q}}^{\ast }$. Since the Fourier transform of a derivative converts ${\nabla }^2$ to $-q^2$, each Fourier mode obeysIt is an Ornstein-Uhlenbeck process. Multiplication by the integrating factor $e^{r(q)t}$ givesThe noise integral is understood as an Itô integral with a deterministic kernel; writing $\eta ds$ is shorthand for the corresponding Wiener increment. Direct differentiation in the mild stochastic sense checks both the equation and the initial value. The stated Kronecker-delta noise covariance corresponds to this finite-volume normalization. Unlike a conserved order parameter, the zero wavenumber mode decays at the nonzero rate $\Gamma a$.

The post-quench Ornstein-Uhlenbeck process has $r_1(q)=\Gamma (a_1+\kappa q^2)$. Its future noise is independent of the initial field. Consequently the cross term between the initial Fourier mode and its future noise integral has zero expectation. The Itô isometry, or the given white-noise covariance, then yieldsThe initial Gaussian Boltzmann distribution, with $a=a_0$, gives $S_q(0)=k_{B}T/(a_0+\kappa q^2)$ in the same unitary normalization. Thus the Gaussian Model A quench hasThe static structure factor interpolates monotonically mode by mode between its two positive equilibrium values. The covariance relaxes at twice the amplitude decay rate. These statements assume that $T$, $\Gamma$ and $\kappa$ are unchanged by the quench; both $a_0,a_1>0$ exclude a spinodal instability.

For a deterministic localized initial peak in an unbounded $d$-dimensional sample, the mean obeys ${\partial }_t⟨\varphi ⟩=\Gamma \kappa {\nabla }^2⟨\varphi ⟩-\Gamma a⟨\varphi ⟩$. The Gaussian Model A impulse response is therefore the heat kernel multiplied by the local relaxation factor:The peak spreads over a distance of order $\sqrt{\Gamma \kappa t}$ and its central height falls both by spreading and by exponential relaxation. Its integrated mean is $A{e}^{-\Gamma at}$, rather than $A$: the equation describes nonconserved order-parameter dynamics, despite the use of the word “density” in this part.

An individual realization also develops thermal fluctuations; it does not tend pointwise to zero. For an initially deterministic field, the connected Fourier mode variance grows as $[k_{B}T/(a+\kappa q^2)](1-e^{-2r(q)t})$, while the deterministic mean decays. It approaches the zero-mean Gaussian equilibrium field at the coarse-graining cutoff. On a periodic or bounded sample use the corresponding boundary-adapted heat kernel; at $\kappa =0$ there is no spatial spreading. The Dirac delta distribution is an ideal initial profile, recovered as a distributional limit; a finite-energy physical peak has a microscopic width.

The field $\varphi$ is a compositional order parameter in a binary fluid mixture. Its spatial integral is conserved in a closed sample, so advection and diffusion enter a continuity equation; the material derivative follows the incompressible fluid. The diffusive current $\mathbf{J}=-M\nabla \mu$, with positive order-parameter mobility $M$, moves material down the chemical potential gradient. The chemical potential of a composition field is the functional derivative of the symmetric quartic Landau free energyso $\mu =a\varphi +b{\varphi }^3-\kappa {\nabla }^2\varphi$. Positive $b,\kappa$ stabilize large amplitudes and penalize sharp interfaces; $a<0$ permits phase separation.

The momentum equation is the Navier-Stokes equation with dynamic viscosity $\eta$, constant mass density $\rho$, and the Korteweg force $-\varphi \nabla \mu$ arising from composition stresses. The pressure enforces incompressible flow, $\nabla \cdot \mathbf{v}=0$, and can absorb pure gradient contributions to that thermodynamic force. For example, $-\varphi \nabla \mu =\mu \nabla \varphi -\nabla (\varphi \mu )$ gives an equivalent force after redefining the pressure. These equations constitute deterministic Model H dynamics.

Their dissipative signs can be checked by the energy dissipation identity for Model H. Under periodic or no-flux, no-work boundary conditions, composition advection and capillary work cancel between $F$ and the kinetic energy, givingThus diffusion and viscosity dissipate energy, while the reversible advective coupling transfers it between composition and motion. Thermal noises are omitted at this deterministic hydrodynamic level.

The dynamical scaling of binary-fluid coarsening assumes a statistically self-similar bicontinuous morphology with one growing length $L$, thin interfaces, curvature of order $1/L$, and typical fluid speed of order $\dot{L}$. Inertial acceleration then scales as $\ddot{L}$ and ${\dot{L}}^2/L$, viscous force density as $\eta \dot{L}/L^2$, and capillary force density as $\sigma /L^2$. The pressure is eliminated or estimated together with the interfacial force. The numerical coefficients summarize geometry and projected force directions; this is a scaling closure, not a spatially exact momentum equation.

The approximation further neglects diffusive growth relative to advective growth, assumes fixed volume fraction and material parameters, no external forcing, and no additional relevant macroscopic length. A finite initial domain size, interface width or finite container would introduce further ratios outside the asymptotic scaling regime. Velocity gradients need not stay on the domain scale in a genuinely multiscale turbulent flow, so this single-length argument is not a proof for every hydrodynamic morphology.

With $[\rho ]=\mathrm{M}{\mathrm{L}}^{-3}$, $[\eta ]=\mathrm{M}{\mathrm{L}}^{-1}{\mathrm{T}}^{-1}$ and $[\sigma ]=\mathrm{M}{\mathrm{T}}^{-2}$, dimensional analysis gives the unique scalesThus, within the stated asymptotic single-scale hypothesis, the hydrodynamic coarsening crossover scales give $L=L_{0}f(u)$, $u=t/t_0$. All four force-density terms have the same dimensional prefactor ${\rho }^2{\sigma }^3/{\eta }^4$. Dividing by it yieldsPrimes denote $d/du$. The PDF includes $\eta \gamma \dot{L}/L^2$; the TeX drops $\eta$. The PDF expression is dimensionally consistent. For a positive growth branch, a resisting viscous term and a driving capillary term have opposite signed coefficients; one may set $\gamma =-c_v<0$, $\delta =c_s>0$. The question calls these coefficients order one, not positive. Treating every term on the printed right-hand side as positive would lose the viscous-capillary balance.

For a power law $f(u)=C{u}^p$, restoring dimensions givesIndependence of mass density requires $p-1=0$. Hence viscous hydrodynamic coarsening hasAt small $u$, the capillary and viscous terms are both of order $u^{-2}$, while the convective inertial term is of order $u^{-1}$ and the acceleration of the leading linear law is zero. Thus the leading force balance is viscous resistance against capillary driving, giving $C_v=c_s/c_v=-\delta /\gamma >0$ in the signed convention of the preceding part.

Equivalently, the domain-scale Reynolds number is $\rho L\dot{L}/\eta \sim C_v^{2}u$, so inertia is negligible as $u\to 0$. “Small” here refers to time compared with $t_0$ within a late-stage hydrodynamic window: it does not extend the model through the microscopic initial stage, where diffusion or interface formation can dominate. An initial-time offset changes the intercept, not this asymptotic growth exponent.

For $f(u)=C{u}^p$, the dimensional expression is again $L=C{\eta }^{2-3p}{\rho }^{p-1}{\sigma }^{2p-1}{t}^p$. Independence of dynamic viscosity requires $2-3p=0$, giving inertial hydrodynamic coarsening:For this law, both inertial terms and the capillary term scale as $u^{-4/3}$, whereas the viscous term scales as $u^{-5/3}$. It is consequently subleading at large $u$. Direct substitution gives the leading amplitude conditionA growing branch requires the indicated ratio to be positive. The inertial combination must be nonzero: if $2\beta =\alpha$ its leading acceleration and convective contributions cancel, so this particular schematic equation cannot realize the generic inertial-capillary balance with that pure power law. The requested scaling assumes no such geometrical degeneracy. This also shows why the coefficients cannot determine a universal numerical prefactor without a morphology calculation. The relative viscous contribution decays as $u^{-1/3}$, consistently with an increasingly large domain-scale Reynolds number.

A stationary microporous network removes fluid momentum through linear drag rather than through a domain-scale viscous stress. With $[\bar{\eta }]=\mathrm{M}{\mathrm{L}}^{-3}{\mathrm{T}}^{-1}$, the single-scale force estimate becomesThis drag-limited hydrodynamic coarsening closure uses a fixed isotropic drag coefficient and domain sizes large compared with the pore scale. The only length and time made from $\rho ,\sigma ,\bar{\eta }$ areIndeed $L_1^3=\sigma t_1^2/\rho$, while the choice of $t_1$ equates inertial and drag prefactors. With $L=L_{1}g(u)$ and $u=t/t_1$, all force terms share $\rho L_1/t_1^2=\bar{\eta }{L}_1/t_1=\sigma /L_1^2$. ThereforeAs before, a parameter-free function of this one argument refers to the asymptotic advective scaling model with fixed morphology and negligible extra scales. Retaining the diffusive mobility or resolved network geometry can introduce additional dimensionless parameters. The original $\eta$ does not remain as an independent viscous coefficient after the stipulated replacement.

Use the intended regular scaling ansatz: $g(u)\sim C{u}^y$, $g^{\prime }(u)\sim Cy{u}^{y-1}$ and $g^{\prime \prime }(u)=Cy(y-1)u^{y-2}+o(u^{y-2})$, with $C>0$, $y>0$. Relative to the drag magnitude $g^{\prime }$, the two inertial contributions obeyThus both inertial-to-drag ratios tend to zero at large $u$, whatever the positive growth exponent; for $y=1$ the leading acceleration term vanishes identically. The long-time drag-limited hydrodynamic coarsening balance is thereforesoThe leading law is independent of mass density. Within the drag-capillary reduction, finite initial data give $L^3(t)-L^3(t_{\ast })=3(c_s/c_d)(\sigma /\bar{\eta })(t-t_{\ast })$.

The derivative hypotheses are important: bare $g(u)\sim u^y$ does not justify differentiating an asymptotic equivalence. For instance, $g(u)=u+\frac{1}{2}{\int }_1^u\sin (s^3)ds$ is smooth and increasing, with $g(u)/u\to 1$, but $g^{\prime \prime }=\frac{3}{2}{u}^2\cos (u^3)$ is not small compared with $g^{\prime }=1+\frac{1}{2}\sin (u^3)$. This is a counterexample to an inference from the bare asymptotic relation, not a solution of the coarsening equation. The physical claim uses the regular self-similar power law, for which the ratio calculation above applies.

This conclusion concerns the model's advective drag-capillary channel. Diffusive Ostwald ripening can also have a $1/3$ growth exponent; equality of exponents does not establish that diffusion is negligible. If it is retained, the prefactor and crossover function may also depend on order-parameter mobility, and one must compare transport mechanisms rather than infer the mechanism from the exponent alone.

A nematic director is an unoriented axis: $\mathbf{n}$ and $-\mathbf{n}$ describe the same molecular alignment. A vector average cannot represent this head-tail symmetry. The orientational second moment $⟨n_i{n}_j⟩$ is a symmetric second-rank tensor, and removing its isotropic part gives the traceless second-rank tensor $Q_{ij}\propto ⟨n_i{n}_j-{\delta }_{ij}/2⟩$. It vanishes for an isotropic angular distribution and transforms correctly under rotation matrices.

Every nonzero real symmetric traceless $2\times 2$ matrix has eigenvalues $\lambda /2,-\lambda /2$ and can be written $Q=\lambda (\mathbf{n}{\mathbf{n}}^{\mathsf{T}}-I/2)$ by choosing the eigenvector of the positive eigenvalue as director and $\lambda \ge 0$. Consequently $\mathrm{Tr}{Q}^2={\lambda }^2/2$. The bulk Landau-de Gennes free energy becomesFor $a<0$, its minimum occurs at ${\lambda }^2=-a/b$. The elastic density $K|\nabla \cdot Q{|}^2/2$ is nonnegative and vanishes for a uniform field. Thus, with no boundary condition imposing a texture, the pointwise lower bound is attained byThe negative-amplitude representation is equivalent in two dimensions to rotating the director by $\pi /2$, so the positive choice loses no tensor minima. This is minimization of the specified coarse-grained functional, not a claim of true thermodynamic long-range nematic order at every temperature in an infinite two-dimensional system. The PDF explicitly uses the divergence elastic term; the TeX's $|\nabla Q{|}^2$ loses the dot and would give different elastic coefficients.

Put $\mathbf{t}=\mathbf{\theta }$. The product rule applied to $Q_{ij}=\lambda (t_i{t}_j-{\delta }_{ij}/2)$ givesSince $\mathrm{Tr}{Q}^2={\lambda }^2/2$, substituting this vector into the divergence elastic term proves the requested form of the local density. For the radial amplitude and tangential director, the polar coordinates identities areHence the tangential integer nematic defect core hasThe $1/r$ director-curvature term survives when the amplitude is constant. Replacing the divergence norm by a full tensor-gradient norm would not give this expression; it is the PDF's divergence convention that is being evaluated.

Outside a core of radius $r_0$, set $\lambda \simeq {\lambda }_0$ to leading order. The uniform-state bulk energy cancels on subtraction, while the divergence elastic density is $K{\lambda }_0^2/(2r^2)$. Its two-dimensional area integral isThus the logarithmic energy of a planar nematic disclination givesThe core suppresses the nematic amplitude to regularize the singular director texture. Balancing lost bulk ordering energy against elasticity gives $r_0$ of order $\sqrt{K/|a|}$. The exact core radius and additive core energy depend on the amplitude profile.

The “local” amplitude disturbance need not be literally compactly supported. The radial Euler-Lagrange equation following from the PDF density isIts far-field expansion is $\lambda ={\lambda }_0[1-K/(2|a|r^2)+O(r^{-4})]$. These algebraic tails change only finite contributions, not the logarithm. A regular core has $\lambda =O(r^2)$ for this integer winding.

The large cutoff $L$ is set by the sample size for an isolated defect with corresponding imposed far-field winding, or by the distance to compensating defects or other texture-screening boundaries. An isolated nonzero winding in an infinite ordered plane has a logarithmically divergent excess energy; in a neutral pair, far-field distortions cancel beyond the separation. If $L$ is not much larger than $r_0$, the leading logarithm is not a dominant approximation.

For a strictly planar nematic director, the order-parameter manifold at fixed nonzero amplitude is ${\mathbb{RP}}^1$, homeomorphic to a circle. Its fundamental group is $\mathbb{Z}$. The topological charge of a two-dimensional nematic disclination is $q=\Delta \chi /(2\pi )$ for director angle $\chi$, defined modulo $\pi$; the integer winding of the physical doubled angle $2\chi$ is $m=2q$. Thus a $q=+1$ texture has $m=2\ne 0$ and cannot be continuously replaced by a uniform nonvanishing planar texture while the surrounding loop is held fixed:It can nevertheless split into two $q=+1/2$ defects, because their charges add to one. Such splitting preserves the total winding and is not topological unwinding. Allowing escape into the third dimension changes the manifold to ${\mathbb{RP}}^2$, with fundamental group $\mathbb{Z}/2\mathbb{Z}$; the doubled winding of this integer defect is then trivial. That escape is excluded by the stated two-dimensional director model.

Energetically, the defect is not the preferred isolated charge-one configuration at large separation of scales. At constant amplitude, the planar nematic divergence-elasticity identity gives $F_{\mathrm{el}}=(K{\lambda }_0^2/2)\int |\nabla \chi {|}^2$, so a charge $q$ has self-energy $\pi K{\lambda }_0^2{q}^2\log (L/r_0)$. Separating two $+1/2$ defects by $d$, with $r_0\ll d\ll L$, makes the far field still charge one but reduces the near-field logarithm:The logarithmic gain eventually dominates the finite core cost. HenceThis is energetic fission of an integer nematic disclination. It proves failure of global energetic stability, not the absence of every microscopic kinetic barrier or a full linear stability spectrum of the radial core. Core structure, boundaries or pinning can make a charge-one texture metastable over a finite time or length scale.

The total charge of the widely separated $+1,-1$ pair is zero, so a uniform planar nematic director is compatible with the far boundary if it imposes no additional winding. Each integer defect is susceptible to energetic fission of an integer nematic disclination: the positive defect can split into two $+1/2$ defects and the negative defect into two $-1/2$ defects. Like-sign elementary defects repel, whereas opposite signs attract through their elastic distortion. This commonly produces subsequent encounters and annihilations of opposite half-integer defects.

The attraction can also be seen before fission. In the fixed-amplitude outer theory, a neutral pair of charges $q,-q$ separated by $R$ hasIts energy therefore falls as the separation decreases. In nematic defect-antidefect annihilation, opposite defects meet, the amplitude can vanish in their merging cores, and the local winding disappears; the remaining smooth director distortion relaxes towards the uniform energy minimum.

Direct approach and annihilation of the unsplit integer pair is also topologically allowed. Whether fission happens first, the paths of the four elementary defects, and the annihilation time require a specified dynamical law and core mobilities; the free energy alone does not fix them. Pinning, imposed boundary textures, or sustained thermal production of defect pairs can prevent a strictly defect-free final state. The stated final state presumes dissipative relaxation with no such obstruction.