Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 344 / 1

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.