From Fluid to Crystal - A Thermodynamic and Dynamical-Systems Note on Nucleation

Posted on 2026-07-26 In Physics , Statistical Mechanics Views: Disqus: Word count in article: 20k Reading time ≈ 1:12

ENGLISH

From Fluid to Crystal

A Thermodynamic and Dynamical-Systems Note on Nucleation

Wanhong Huang

Abstract

This note develops the passage from a metastable fluid to a crystalline solid in several formal registers. Sections 1–3 treat the transition as a variational problem on a free-energy landscape: the critical nucleus is characterised as a saddle point of the landscape, and the activation barrier is derived. Section 4 reformulates the same content as a one-dimensional dynamical system, in which the critical radius appears as an unstable fixed point (a repeller) separating two basins of attraction. Sections 5–7 restore the stochastic element that both deterministic pictures suppress, and show that the registers are reconciled by a first-passage argument. Sections 8–9 then mark the limit of that apparatus: when growth becomes diffusion-limited the capillarity assumptions fail, the interface goes unstable, and the resulting aggregate acquires a non-integer dimension fixed by its own growth history, with no energy minimum selecting it. Section 10 then derives that regime’s structure: self-similarity forces a power law, screening of the diffusion field supplies the mechanism, and a mean-field argument estimates the exponent. Throughout, the emphasis falls on why the transition is threshold-governed while remaining continuous, and on the precise sense in which a small ordered configuration can reorganise its environment. All figures are produced by direct simulation of the equations stated in the text.

Keywords: classical nucleation theory; critical nucleus and free-energy barrier; unstable fixed point and separatrix; Kramers barrier crossing and first-passage time; diffusion-limited aggregation and fractal dimension.

Introduction and reading guide

The elementary account of freezing says that below the melting point the crystal is the lower-energy state, and therefore the fluid freezes. This is incomplete in an important way: it explains the destination while leaving the route unaccounted for. A supercooled liquid can persist for a long time in a state above the global minimum of its free energy. Something must be paid before the favourable state can be reached, and the currency is interface.

The whole of classical nucleation theory follows from one asymmetry: bulk energy scales with volume, interfacial energy scales with area. Since $r^{3}$ eventually beats $r^{2}$ but loses to it for small $r$, there is a range of sizes in which order is penalised and a range in which it is rewarded. The boundary between them is a genuine threshold, and this is what gives the phenomenon its mathematical interest beyond mere description.

Sections 1–3 derive that threshold thermodynamically. Section 4 shows it is the same object as an unstable equilibrium of a flow. Sections 5–7 explain what actually gets a system across it. Sections 8–9 ask what the resulting crystal looks like, and find that the preceding framework leaves this question entirely unanswerable. Section 10 supplies the missing account.

§1 The Metastable State

Definition 1.1 (Phases and order parameter). Let a one-component system be held at fixed temperature $T$ and pressure $p$. Its equilibrium states are the minima of the Gibbs free energy $G = U - TS + pV$. We write $\varphi$ for an order parameter: a scalar field distinguishing the two candidate phases, with $\varphi \approx 0$ in the disordered (fluid) phase and $\varphi \approx 1$ in the ordered (crystalline) phase. Physically $\varphi$ may be taken as a local measure of translational and orientational periodicity.

Definition 1.2 (Metastability). A state $\varphi_{0}$ is metastable if it is a local but not global minimum of $G$. Formally: there is a neighbourhood $\mathcal{U} \ni \varphi_{0}$ with $G(\varphi_{0}) \le G(\varphi)$ for all $\varphi \in \mathcal{U}$, while there exists $\varphi_{1} \notin \mathcal{U}$ with $G(\varphi_{1}) < G(\varphi_{0})$.

Remark 1.3. Definition 1.2 is the formal content of the phrase “the liquid has not yet transformed.” Metastability is a statement about the shape of $G$ between the two minima: the liquid is protected by the fact that every path to the crystal passes through states of higher free energy. Supercooled water at $-10^{\circ}\mathrm{C}$ is stable against all small perturbations and unstable only against sufficiently large ones. Everything below is an attempt to make “sufficiently large” precise.

Definition 1.4 (Undercooling and the bulk driving force). Let $T_{m}$ be the melting temperature and let $\Delta T := T_{m} - T > 0$ denote the undercooling. Define the bulk free-energy density difference
$$\Delta g_v ;:=; \frac{G_{\mathrm{liquid}} - G_{\mathrm{crystal}}}{V} ;>; 0 ,$$
the free energy released per unit volume of fluid converted to crystal.

Proposition 1.5 (Linearisation of the driving force near $T_{m}$). To first order in $\Delta T$,
$$\Delta g_v ;\simeq; \frac{L,\Delta T}{T_{m}} ,$$
where $L$ is the latent heat of fusion per unit volume.

Proof. Write $\Delta g := g_{\ell} - g_{c} = \Delta h - T,\Delta s$ per unit volume. At coexistence $T = T_{m}$ we have $\Delta g = 0$, hence $\Delta s = \Delta h / T_{m} = L/T_{m}$. Treating $\Delta h \simeq L$ and $\Delta s \simeq L/T_{m}$ as temperature-independent over the small interval $[T, T_{m}]$,
$$\Delta g(T) ;\simeq; L - T\frac{L}{T_{m}} ;=; \frac{L,(T_{m}-T)}{T_{m}} ;=; \frac{L,\Delta T}{T_{m}} . \qquad\blacksquare$$

Remark 1.6. The proposition records that $\Delta g_v$ vanishes linearly at the melting point. Every quantity derived below depends on $\Delta g_v$ in the denominator, so all of them diverge as $T \to T_{m}^{-}$: at coexistence there is no driving force and no nucleation. Depth of undercooling is the single control parameter of the theory.

§2 The Free-Energy Functional of a Nucleus

Assumption 2.1 (Capillarity approximation). We adopt the standard idealisations of classical nucleation theory:

(A1) Sharp interface. The nucleus is a region of pure crystal separated from pure fluid by a mathematically sharp surface of zero thickness.

(A2) Bulk properties. The interior of the nucleus has the bulk crystal free-energy density, however small the nucleus.

(A3) Isotropic, size-independent interfacial tension. The surface costs $\gamma > 0$ per unit area, independent of $r$ and of orientation.

(A4) Sphericity. Consequently, by (A3) and the isoperimetric inequality, the minimising shape at fixed volume is a sphere, parametrised by its radius $r \ge 0$.

Remark 2.2. These assumptions are quantitatively poor for real crystals, (A3) fails badly for faceted materials, and (A2) is dubious for nuclei of ten molecules, yet the structure they produce (a barrier, a critical size, an exponential rate law) is robust and survives in far more sophisticated treatments. This note is about that structure.

Definition 2.3 (Nucleation free energy). Under Assumption 2.1, the free-energy cost of forming a spherical crystal nucleus of radius $r$ inside the fluid is the function $\Delta G \colon [0,\infty) \to \mathbb{R}$,
$$\boxed{;\Delta G(r) ;=; \underbrace{4\pi r^{2}\gamma}{\text{interfacial cost}} ;-; \underbrace{\tfrac{4}{3}\pi r^{3},\Delta g_v}{\text{bulk gain}} ; } \tag{1}$$

Remark 2.4 (The structural asymmetry). Equation (1) is a difference of two monomials of different degree, with the lower degree carrying the positive sign. This single fact generates everything that follows. Writing $\Delta G(r) = ar^{2} - br^{3}$ with $a,b>0$: as $r \to 0^{+}$ the quadratic dominates, so $\Delta G$ increases; as $r \to \infty$ the cubic dominates, so $\Delta G \to -\infty$. A continuous function that first rises and then falls without bound must possess an interior maximum. The barrier follows from the mismatch of scaling exponents alone, with no additional physical hypothesis required.

§3 The Critical Nucleus as a Saddle Point

Theorem 3.1 (Critical radius and activation barrier). Let $\Delta G$ be as in (1) with $\gamma > 0$ and $\Delta g_v > 0$. Then $\Delta G$ has exactly one interior stationary point on $(0,\infty)$, located at
$$r_{c} ;=; \frac{2\gamma}{\Delta g_v} , \tag{2}$$
this point is a strict local maximum, and the corresponding value is
$$\Delta G^{*} ;:=; \Delta G(r_{c}) ;=; \frac{16\pi\gamma^{3}}{3,\Delta g_v^{2}} ;>; 0 . \tag{3}$$
Moreover $\Delta G$ is strictly increasing on $(0,r_{c})$, strictly decreasing on $(r_{c},\infty)$, and vanishes at the single positive root $r_{0} = \tfrac{3}{2} r_{c}$.

Proof. Differentiating (1),
$$\Delta G’(r) ;=; 8\pi r \gamma - 4\pi r^{2}\Delta g_v ;=; 4\pi r,(2\gamma - r,\Delta g_v).$$
On $(0,\infty)$ the factor $4\pi r$ is strictly positive, so $\Delta G’(r)=0$ iff $r = r_{c} = 2\gamma/\Delta g_v$, establishing uniqueness. The sign of $\Delta G’$ is that of $(2\gamma - r\Delta g_v)$, which is positive for $r<r_{c}$ and negative for $r>r_{c}$; this gives the monotonicity claims and shows $r_{c}$ is a strict maximum. For the second-derivative confirmation,
$$\Delta G’’(r) = 8\pi\gamma - 8\pi r \Delta g_v, \qquad \Delta G’’(r_{c}) = 8\pi\gamma - 8\pi\frac{2\gamma}{\Delta g_v}\Delta g_v = -8\pi\gamma < 0 .$$
Substituting (2) into (1),
$$\Delta G^{*} = 4\pi\gamma\frac{4\gamma^{2}}{\Delta g_v^{2}} - \frac{4\pi}{3}\frac{8\gamma^{3}}{\Delta g_v^{3}}\Delta g_v = \frac{16\pi\gamma^{3}}{\Delta g_v^{2}} - \frac{32\pi\gamma^{3}}{3\Delta g_v^{2}} = \frac{16\pi\gamma^{3}}{3\Delta g_v^{2}} .$$
Finally $\Delta G(r) = 4\pi r^{2}(\gamma - \tfrac{1}{3} r \Delta g_v)$ vanishes for $r>0$ iff $r = 3\gamma/\Delta g_v = \tfrac{3}{2}r_{c}$. $\blacksquare$

Corollary 3.2 (Scaling with undercooling). Using Proposition 1.5,
$$r_{c} ;\propto; \frac{1}{\Delta T}, \qquad \Delta G^{} ;\propto; \frac{1}{(\Delta T)^{2}} .$$
Hence $r_{c}, \Delta G^{} \to \infty$ as $\Delta T \to 0^{+}$, and both decrease monotonically as the melt is cooled further below $T_m$.

Remark 3.3 (The critical nucleus as a transition state). Theorem 3.1 states that the critical nucleus sits at a maximum of $\Delta G$ along the radial coordinate. In the full configuration space, where the state also has many other degrees of freedom, transverse to the reaction coordinate $r$, along which $\Delta G$ is minimised, this point is a first-order saddle: a maximum in exactly one direction and a minimum in all others. This is the precise sense in which the critical nucleus is a transition state: the system passes through it without resting there, and it is the least-cost gateway between the two states where rest is possible.

The consequence deserves emphasis, because it inverts ordinary intuition about stability. The critical nucleus is the configuration that is maximally unstable along its own growth direction. Anything smaller decays; anything larger runs away; the critical object itself is poised exactly on the divide. Order at the threshold is the most precarious form of order, and the least robust.

Example 3.4 (Orders of magnitude). For water at $\Delta T = 10,\mathrm{K}$, with $\gamma \approx 30,\mathrm{mJ,m^{-2}}$ and $L \approx 3.0 \times 10^{8},\mathrm{J,m^{-3}}$, Proposition 1.5 gives $\Delta g_v \approx 1.1 \times 10^{7},\mathrm{J,m^{-3}}$, whence
$$r_{c} \approx \frac{2(0.030)}{1.1\times 10^{7}} \approx 5.5,\mathrm{nm},$$
a nucleus containing on the order of $10^{4}$ molecules. The corresponding barrier $\Delta G^{*}$ is some tens of $k_{\mathrm B} T$, large enough that spontaneous crossing is rare, which is exactly why supercooled water exists.

Figure 1. (a) The landscape $\Delta G(r)$ of Definition 2.3 at three driving forces, with the maximum of Theorem 3.1 marked. Deeper undercooling (larger $\Delta g_v$) lowers and left-shifts the barrier, as Corollary 3.2 requires. (b) Decomposition into the two competing terms. The barrier follows from $r^{2}$ dominating $r^{3}$ at small $r$ and losing at large $r$ (Remark 2.4).

§4 Gradient Flow on the Landscape

Sections 1–3 described a static object: a curve. We now ask how a nucleus moves on that curve. The simplest physically defensible postulate is that the system descends the free energy at a rate proportional to the local slope.

Assumption 4.1 (Overdamped gradient dynamics). The radius evolves according to
$$\frac{dr}{dt} ;=; -M,\frac{d,\Delta G}{dr}(r), \qquad M > 0, \tag{4}$$
where $M$ is a mobility (kinetic prefactor) encoding how quickly molecules can attach to or detach from the interface. Inertia is neglected: this is the overdamped limit appropriate to condensed matter.

Definition 4.2 (Fixed point, stability). For a flow $\dot{r} = f(r)$ on $\mathbb{R}$, a point $r^{}$ with $f(r^{}) = 0$ is a fixed point (equilibrium). It is linearly stable (an attractor) if $f’(r^{*}) < 0$, and *linearly unstable* (a repeller) if $f’(r^{*}) > 0$.

Proposition 4.3 (Explicit vector field). Under Assumptions 2.1 and 4.1,
$$\frac{dr}{dt} ;=; f(r) ;:=; -4\pi M, r,\bigl(2\gamma - r,\Delta g_v\bigr) ;=; 4\pi M \Delta g_v, r,(r - r_{c}). \tag{5}$$

Proof. Immediate from (4) and the expression for $\Delta G’$ computed in the proof of Theorem 3.1, using $r_{c} = 2\gamma/\Delta g_v$. $\blacksquare$

Theorem 4.4 (Phase portrait of nucleation). The flow (5) on $[0,\infty)$ has exactly two fixed points:

$r^{*}{1} = 0$, which is stable (an attractor), with $f’(0) = -4\pi M \Delta g_v, r{c} = -8\pi M \gamma < 0$;

$r^{*}{2} = r{c}$, which is unstable (a repeller), with $f’(r_{c}) = 4\pi M \Delta g_v, r_{c} = 8\pi M \gamma > 0$.

Consequently $\mathbb{R}{\ge 0}$ decomposes into two invariant intervals:
$$\underbrace{[0, r{c})}{\text{basin of attraction of }0} ;;\cup;; {r{c}} ;;\cup;; \underbrace{(r_{c}, \infty)}{\text{escape to unbounded growth}} .$$
For $r(0) \in [0,r{c})$ the solution satisfies $r(t) \to 0$ as $t \to \infty$ (dissolution). For $r(0) \in (r_{c},\infty)$ the solution is strictly increasing and unbounded (growth).

Proof. By (5), $f(r) = 4\pi M \Delta g_v, r(r-r_{c})$ vanishes exactly at $r=0$ and $r=r_{c}$. Differentiating, $f’(r) = 4\pi M \Delta g_v,(2r - r_{c})$, and evaluating at the two fixed points gives the stated signs, which classify them by Definition 4.2. On $(0,r_{c})$ the product $r(r-r_{c})$ is negative, so $\dot{r} < 0$ and $r$ decreases monotonically; being bounded below by $0$ it converges, necessarily to a fixed point, hence to $0$. On $(r_{c},\infty)$ the product is positive, so $r$ increases monotonically; it cannot converge to a finite limit since there is no fixed point above $r_{c}$, hence $r(t) \to \infty$. $\blacksquare$

Figure 2 (phase line). The phase line of Theorem 4.4. Filled circle: the attractor at $r=0$. Open circle: the repeller at $r_c$. Arrows give the sign of $\dot r$ ($\dot r < 0$, dissolution / negative feedback, on $(0,r_c)$; $\dot r > 0$, growth / positive feedback, on $(r_c,\infty)$), so the two basins carry opposite flow.

Corollary 4.5 (Feedback signs). Write $\delta(t) = r(t) - r^{}$ for a small perturbation about a fixed point. Then $\dot{\delta} \simeq f’(r^{}),\delta$, so
$$\delta(t) \simeq \delta(0),e^{f’(r^{})t}.$$
At $r^{}{1}=0$ the exponent is negative: perturbations are damped, which is the formal content of “negative feedback dissolves small nuclei.” At $r^{*}{2}=r_{c}$ the exponent is positive: perturbations are amplified, which is the formal content of “positive feedback drives runaway growth.” Both statements are one linearisation of the same vector field at two different points.

Remark 4.6 (The repeller as a separatrix). In one dimension an unstable fixed point is precisely a separatrix: it partitions the state space into basins whose futures are qualitatively incomparable. Two initial conditions differing by an arbitrarily small $\epsilon$ across $r_{c}$ have divergent long-time fates, one vanishes, one grows without bound. This sensitivity is localised entirely at the threshold; away from it, the dynamics is dull and predictable.

Note that the same point carries two different names in the two registers. Along the reaction coordinate, the maximum of $\Delta G$ (Theorem 3.1) is the repeller of the flow (Theorem 4.4). The two are related by an identity: for a gradient flow $\dot r = -M,\Delta G’(r)$ one has $f’(r) = -M,\Delta G’’(r)$, so the stability of a fixed point is the sign-reversed curvature of the landscape. Maxima repel, minima attract. The two formalisms are the same statement read in two directions.

Remark 4.7 (On the bifurcation-theoretic reading). It is common to describe nucleation as a subcritical bifurcation, and the analogy is useful, but a caution is in order. Strictly, a bifurcation is a change in the number or stability of fixed points as a control parameter is varied. Here the natural control parameter is the undercooling $\Delta T$. From (5) and Corollary 3.2, as $\Delta T \to 0^{+}$ we have $r_{c} \to \infty$: the repeller recedes to infinity and the fluid becomes globally stable. As $\Delta T$ increases, $r_{c}$ descends toward molecular dimensions, and once $r_{c}$ approaches the size of a single structural unit the barrier ceases to be meaningful, the system becomes unstable to any fluctuation. That limit is the spinodal, and it is there, not at finite undercooling, that a genuine loss of local stability occurs.

For fixed $\Delta T$, then, what we have exhibited is a bistable structure with a threshold: two basins separated by a repeller, with no bifurcation involved. The distinction matters. A bifurcation changes what states exist; a threshold governs which of the existing states is reached.

Figure 3. (a) The vector field $\dot r=-M,\Delta G’(r)$ of Proposition 4.3. Shading marks the sign of $\dot r$; the filled circle is the attractor at $r=0$, the open circle the repeller at $r_c$. (b) Trajectories of the same flow. Initial conditions differing by $\pm 0.03$ about $r_c$ separate to opposite fates, which is the separatrix property of Remark 4.6.

§5 Limits of the Deterministic Description

Remark 5.1 (The paradox). Theorem 4.4 contains an embarrassment. The fluid state $r=0$ is a stable attractor: every initial condition below $r_{c}$ returns to it. Taken literally, the deterministic flow (5) predicts that a supercooled liquid never freezes, since it starts at $r=0$ and stays there.

That prediction is false, and its falsity locates the missing ingredient exactly. Equation (4) describes the mean drift of the radius but discards the thermal agitation from which nuclei arise in the first place. Deterministic dynamics can explain what happens to a nucleus once it exists; it cannot explain the existence of a nucleus. Order arises from the fluctuations around the systematic tendency of the system; that tendency itself points the other way.

§6 Langevin and Fokker–Planck Formulations

Definition 6.1 (Langevin equation for the nucleus radius). Restore the thermal term:
$$\frac{dr}{dt} ;=; -M,\frac{d,\Delta G}{dr}(r) ;+; \eta(t), \tag{6}$$
where $\eta$ is Gaussian white noise with
$$\langle \eta(t)\rangle = 0, \qquad \langle \eta(t)\eta(t’)\rangle = 2D,\delta(t-t’),$$
and the diffusion coefficient $D$ is tied to the mobility by the fluctuation–dissipation relation $D = M k_{\mathrm B} T$.

Remark 6.2. The relation $D = M k_{\mathrm B} T$ carries physical content beyond modelling convenience. It states that the same molecular collisions that impede growth (dissipation, $M$) are the ones that generate fluctuations ($D$). One cannot have a system that resists motion but does not jitter; a medium capable of destroying a nucleus is thereby capable of creating one. The mechanism of destruction and the mechanism of creation are the same mechanism.

Definition 6.3 (Fokker–Planck equation). Let $P(r,t)$ be the probability density of finding a nucleus of radius $r$ at time $t$. Corresponding to (6), $P$ obeys
$$\frac{\partial P}{\partial t} ;=; \frac{\partial}{\partial r}!\left[ M,\frac{d,\Delta G}{dr},P ;+; D,\frac{\partial P}{\partial r} \right] ;=; -\frac{\partial J}{\partial r}, \tag{7}$$
where
$$J(r,t) ;=; -M\frac{d,\Delta G}{dr}P - D\frac{\partial P}{\partial r} \tag{8}$$
is the probability current in size space.

Remark 6.4 (Reading equation (7)). The bracket in (7) contains the two competing tendencies of the whole theory in adjacent terms. The first is drift: deterministic sliding down the landscape, which for $r<r_{c}$ points toward dissolution. The second is diffusion: spreading of probability irrespective of energetics, which pushes some density uphill. Nucleation is the residual of this competition, the small leakage of probability that diffusion carries over a barrier that drift is trying to enforce.

Proposition 6.5 (Equilibrium distribution). The stationary solution of (7) with zero current, $J \equiv 0$, is the Boltzmann distribution
$$P_{\mathrm{eq}}(r) ;=; \mathcal{N}\exp!\left(-\frac{\Delta G(r)}{k_{\mathrm B} T}\right).$$

Proof. Setting $J=0$ in (8) gives $D,P’ = -M,\Delta G’,P$, i.e. $(\ln P)’ = -(M/D),\Delta G’ = -\Delta G’/k_{\mathrm B} T$ using $D = M k_{\mathrm B} T$. Integrating yields the stated form. $\blacksquare$

Remark 6.6. Proposition 6.5 already contains the essential exponential. It says that subcritical nuclei remain rare yet always present, populated with weight $e^{-\Delta G(r)/k_{\mathrm B} T}$. A supercooled liquid at any instant contains a swarm of small transient ordered clusters, almost all of which dissolve. Nucleation is the occasional survival of order that is being created and destroyed continually.

§7 Nucleation Rate as a First-Passage Problem

Definition 7.1 (Nucleation rate). The steady-state nucleation rate $J_{\mathrm{nuc}}$ is the stationary probability current in (8) under the boundary conditions of a source at small $r$ (maintaining near-equilibrium population of subcritical clusters) and an absorbing sink at large $r$ (supercritical nuclei are removed to macroscopic growth). Equivalently, it is the reciprocal mean first-passage time from $r \approx 0$ across $r_{c}$.

Theorem 7.2 (Kramers/Arrhenius form). For $\Delta G^{} \gg k_{\mathrm B} T$, the steady-state rate obeys
$$\boxed{; J_{\mathrm{nuc}} ;=; A \exp!\left(-\frac{\Delta G^{}}{k_{\mathrm B} T}\right) ;=; A \exp!\left(-\frac{16\pi\gamma^{3}}{3,\Delta g_v^{2},k_{\mathrm B} T}\right) ; } \tag{9}$$
with a kinetic prefactor $A$ determined by the mobility and by the curvature of $\Delta G$ at the two relevant points.

Proof sketch. Impose a constant current $J$ in (8) and solve the resulting first-order linear ODE by the integrating factor $e^{\Delta G(r)/k_{\mathrm B} T}$, giving
$$J \int_{0}^{R} \frac{e^{\Delta G(r)/k_{\mathrm B} T}}{D},dr ;=; P(0)e^{\Delta G(0)/k_{\mathrm B} T} - P(R)e^{\Delta G(R)/k_{\mathrm B} T}.$$
With an absorbing condition $P(R)\to 0$ and $P(0)$ set by local equilibrium, the integral on the left is dominated, for $\Delta G^{}\gg k_{\mathrm B} T$, by the neighbourhood of the maximum $r_{c}$. Laplace’s method there, expanding $\Delta G(r) \simeq \Delta G^{} + \tfrac12 \Delta G’’(r_{c})(r-r_{c})^{2}$ with $\Delta G’’(r_{c}) = -8\pi\gamma < 0$, produces the factor $e^{\Delta G^{*}/k_{\mathrm B} T}$ times a Gaussian width, and inverting gives (9). $\blacksquare$

Corollary 7.3 (Extreme sensitivity to undercooling). Combining (9) with Corollary 3.2,
$$J_{\mathrm{nuc}} ;\propto; \exp!\left(-\frac{C}{T,(\Delta T)^{2}}\right)$$
for a material constant $C$. The rate therefore increases in $\Delta T$ with doubly exponential sharpness: over a narrow window of undercooling, $J_{\mathrm{nuc}}$ can change by many orders of magnitude. This is why freezing appears to have a well-defined onset temperature despite being, formally, a continuous function of $T$.

Figure 4. (a) The deterministic flow started at $r=0.05$: it returns to $0$ and stays there forever, the paradox of Remark 5.1. (b) Twelve Langevin trajectories, Definition 6.1, from the same initial condition; those that reach the absorbing boundary are drawn in red. Noise, not drift, produces the transition. (c) The Boltzmann population of Proposition 6.5: subcritical clusters are exponentially rare but never absent. (d) Crossing rate against $1/k_BT$ from $120$ trajectories per temperature. The slope estimates an effective barrier of $13.5$ against the theoretical $\Delta G^{}=16.8$; the gap is expected, since Theorem 7.2 is asymptotic in $\Delta G^{}/k_BT$ and the prefactor $A$ is itself weakly temperature-dependent.

§8 Scope of the Capillarity Model

Remark 8.1 (A boundary of the model). Everything established so far concerns a single scalar, the radius $r$, and answers a single question: does the nucleus survive? Assumption 2.1 purchased this simplicity at a price that must now be paid. Conditions (A3) and (A4), isotropic $\gamma$, spherical shape, go beyond simplifying the geometry: they presuppose the answer to the question of form. A one-parameter model cannot produce a shape, because it has already assumed one.

Consequently the transition from Sections 1–7 to what follows is a change of regime, distinct from a continuation of the same model into a later time. The fractal morphologies below arise precisely where (A3)–(A4) fail, and the analytical task is to identify what makes them fail.

Definition 8.2 (Two rate-limiting regimes). Let $\tau_{D}$ be the characteristic time for a molecule to diffuse to the growth front, and $\tau_{A}$ the characteristic time for it to attach once it has arrived. Define the regimes:

(R1) Attachment-limited ($\tau_{A} \gg \tau_{D}$): transport is fast, the concentration field around the crystal is nearly uniform, and a molecule that fails to attach at one site samples many others. Growth is governed by interfacial energetics.

(R2) Diffusion-limited ($\tau_{D} \gg \tau_{A}$): transport is slow, the molecule attaches essentially where it first arrives, and the concentration field develops steep gradients around protrusions.

Proposition 8.3 (Regime (R2) breaks the capillarity assumption). In regime (R2) the assumption of a size-independent, shape-neutral interfacial penalty (A3) ceases to control the morphology. Control passes to the diffusion field, which is a nonlocal functional of the entire existing shape.

Argument. Let $u$ be the concentration of growth units, satisfying the quasi-static diffusion problem $\nabla^{2}u = 0$ outside the crystal with $u$ fixed at the interface and at infinity. The local growth velocity is $v_{n} \propto \partial u/\partial n$. For a harmonic function, the normal gradient is largest where the boundary is most convex: protrusions intercept a disproportionate share of the incoming flux, while concavities are screened. Hence a protrusion of height $h$ grows at a rate increasing in $h$.

This is a positive feedback on shape, structurally parallel to the positive feedback on size in Theorem 4.4, and with the same consequence: perturbations above a threshold undergo amplification. Interfacial tension opposes it, since curvature raises the local equilibrium concentration by the Gibbs–Thomson effect, and this penalises exactly the sharpest features. The competition therefore selects a finite wavelength, excluding both the smooth sphere and the infinitely fine dendrite. When the diffusive driving force is large, the band of unstable wavelengths is wide, the selected features are fine, and (A4) is violated at every scale at once. $\blacksquare$

Remark 8.4 (The Mullins–Sekerka instability). Proposition 8.3 is a qualitative statement of a classical linear stability result. Perturbing a growing interface by a mode of wavenumber $k$ and amplitude $\delta_{k}$ gives, schematically,
$$\frac{1}{\delta_{k}}\frac{d\delta_{k}}{dt} ;=; \omega(k) ;\sim; \underbrace{v,k}{\text{diffusive destabilisation}} ;-; \underbrace{\Gamma k^{3}}{\text{capillary stabilisation}},$$
with $\Gamma$ proportional to $\gamma$. The two terms carry different powers of $k$, the same structural device as Remark 2.4, now in wavenumber space with $k$ in place of $r$. Short wavelengths are stabilised by surface tension ($k^{3}$ dominates), long wavelengths are destabilised by the diffusion field ($vk$ dominates), and $\omega(k)$ changes sign at $k^{*} = \sqrt{v/\Gamma}$. Figure 6(e) plots $\omega$ for two driving forces.

Note the recurrence of the pattern: in both cases a stabilising mechanism with a higher power loses to a destabilising mechanism with a lower power in one limit and wins in the other, and the crossover occurs at a threshold, with no gradual trade-off between them.

§9 Diffusion-Limited Aggregation and Fractal Dimension

Definition 9.1 (DLA). Diffusion-limited aggregation is the minimal stochastic model of regime (R2). A seed particle is fixed at the origin. Particles are released one at a time from a distant boundary, perform random walks, and become permanently part of the cluster upon contacting it. A sticking probability $s \in (0,1]$ may be imposed: on contact the walker attaches with probability $s$ and otherwise continues walking.

Remark 9.2 (Why $s$ interpolates between the regimes). The parameter $s$ is the model’s control on Definition 8.2. When $s = 1$ the walker attaches at first contact, necessarily on the outer envelope: this is regime (R2). When $s \ll 1$ the walker rejects most contacts and continues to diffuse, so it penetrates the fjords between branches and fills them: shape-independence is restored and the model approaches regime (R1). One parameter therefore spans both regimes, and the morphological consequence can be read off directly.

Definition 9.3 (Box-counting dimension). For a bounded set $S \subset \mathbb{R}^{2}$, let $N(\epsilon)$ be the number of boxes of side $\epsilon$ in a covering grid that intersect $S$. The box-counting dimension is
$$D ;=; \lim_{\epsilon \to 0} \frac{\log N(\epsilon)}{\log(1/\epsilon)},$$
estimated in practice as the slope of $\log N$ against $\log(1/\epsilon)$ over a range of scales.

Definition 9.4 (Mass–radius exponent). Let $M(R)$ be the number of particles within distance $R$ of the seed. If $M(R) \sim R^{D_{M}}$ over a range of $R$, the exponent $D_{M}$ is the mass dimension. For a compact planar object $D_{M} = 2$; a value $D_{M} < 2$ indicates that the structure fails to fill the plane at any scale.

Remark 9.5 (Numerical results). Figure 5 shows the growth history of a diffusion-limited cluster of $9000$ particles, with colour encoding attachment order. Figure 6 compares the two regimes. The measured values are

box-counting $D$ mass exponent $D_{M}$

diffusion-limited $(s=1)$ 1.54 1.56

attachment-limited $(s=0.03)$ 1.73 1.83

Two observations. First, the two independent estimators agree within each regime, which serves as a consistency check on the measurement, supplying no additional result. Second, and substantively, the diffusion-limited cluster has $D$ strictly between $1$ and $2$: it is neither a curve nor a filled region. The attachment-limited cluster is markedly denser and trends toward the space-filling value $2$.

For honesty about accuracy: the accepted asymptotic value for two-dimensional DLA is $D \approx 1.71$, and the value $1.54$ obtained here is depressed by finite-size effects, since at $9000$ particles the scaling window spans less than two decades. The figure demonstrates the existence of a non-integer dimension and the direction of the regime difference; its accuracy falls short of a precision measurement of the DLA exponent.

	box-counting $D$	mass exponent $D_{M}$
diffusion-limited $(s=1)$	1.54	1.56
attachment-limited $(s=0.03)$	1.73	1.83

Figure 5. Diffusion-limited growth from a single seed ($s=1$), at four stages. Colour encodes attachment order (dark = early). The branched form is present at $N \approx 300$ and is preserved, not outgrown, as the cluster enlarges: later particles extend existing tips while leaving the interior unfilled. No branching rule is imposed; the morphology follows from the screening argument of Proposition 8.3.

Figure 6. The two regimes of Definition 8.2. (a) Diffusion-limited ($s=1$): ramified, with deep fjords that later particles never reach. (b) Attachment-limited ($s=0.03$): the same algorithm with rejection, producing a compact cluster, the regime in which Assumption 2.1 is approximately valid. (c) Box-counting estimates of $D$. (d) Mass–radius scaling, with the compact reference $M \sim R^{2}$ shown dashed. (e) The stability spectrum $\omega(k)$ of Remark 8.4 at two driving forces; the band of unstable modes $\omega>0$ widens with the driving force.

Proposition 9.6 (Fractality as frozen history). In regime (R2) the morphology fails to minimise any free-energy functional. It records the order in which material arrived.

Argument. Attachment in DLA is irreversible: a particle that joins the cluster never detaches, so the system does not sample configurations and cannot relax toward a minimum. The screening of Proposition 8.3 then means each attachment alters the flux field for all subsequent attachments, so early events constrain late ones asymmetrically. Figure 5 exhibits this directly: the dark (early) core retains its shape throughout, and the interior voids visible at $N=300$ are still voids at $N=9000$, because they were screened from the moment they formed. The final object is thus a path-dependent outcome, and no variational principle over final states can select it. $\blacksquare$

Remark 9.7 (Consequence for the equilibrium picture). Proposition 9.6 marks the genuine limit of the thermodynamic register. Sections 1–3 could compute a critical size because that quantity is determined by a competition of energies at a single instant, with no reference to history. Morphology in regime (R2) has no such characterisation. This is why snowflakes are individually distinct while all sharing sixfold symmetry: the symmetry is energetic and therefore universal, while the branch structure is historical and therefore particular to one trajectory through one fluctuating environment.

§10 Theory of Fractal Aggregate Formation

Section 9 measured a dimension. This section derives why a dimension of that kind exists at all, why it lies strictly between $1$ and $2$, and why it is reproducible while the individual branches are not.

10.1 Self-similarity and non-integer dimension

Definition 10.1 (Statistical self-similarity). A random set $S$ is statistically self-similar over a range of scales $[\ell_{\min}, \ell_{\max}]$ if, for $\lambda$ in that range, the rescaled set $\lambda S$ has the same statistical distribution as $S$ up to a change of overall size. The requirement is on the ensemble: individual realisations differ, while their statistics are scale-invariant.

Proposition 10.2 (Self-similarity forces a power law). Let $M(R)$ be the mass within radius $R$ of the seed. If the aggregate is statistically self-similar in the sense of Definition 10.1, then $M$ obeys a power law $M(R) = c R^{D}$ for some exponent $D$, and $D$ is the mass dimension of Definition 9.4.

Proof. Self-similarity means the expected mass profile satisfies $M(\lambda R) = g(\lambda)M(R)$ for all admissible $\lambda, R$, with $g$ depending on $\lambda$ alone. Fix $R_{0}$ and put $\phi(u) = M(R_{0}e^{u})/M(R_{0})$. The relation becomes $\phi(u+v) = \phi(u)\phi(v)$, the Cauchy exponential equation. With $\phi$ measurable and positive, its solutions are $\phi(u) = e^{Du}$ for a constant $D$. Substituting back, $M(R) = M(R_{0})(R/R_{0})^{D}$. $\blacksquare$

Remark 10.3 (What the exponent means). Proposition 10.2 explains why the log–log plots of Figure 6(d) are straight: a straight line there is the statement of scale invariance, and its slope is the only free parameter left. The two integer cases bracket the possibilities. A smooth curve gives $D=1$; a filled region gives $D=2$. A value strictly between them describes an object that becomes sparse as it grows: since the mean density inside radius $R$ scales as
$$\rho(R) ;=; \frac{M(R)}{\pi R^{2}} ;\propto; R^{D-2} ;\xrightarrow[R \to \infty]{} 0 \qquad \text{whenever } D < 2,$$
an infinite DLA cluster has vanishing density in the plane. The aggregate occupies more space than a curve while filling a vanishing fraction of the area available to it, and the exponent $D$ quantifies exactly where between those two limits it falls.

10.2 The screening mechanism

Proposition 10.4 (Screening produces scale-invariant sparsity). In regime (R2) the growth probability concentrates on the outer extremities of the cluster at every stage of growth. Consequently the interior remains permanently underfilled, and the deficit is reproduced at each scale.

Argument. Let $u$ solve the exterior Laplace problem of Proposition 8.3, with the cluster held at fixed potential and a source at infinity. The probability that a random walker from infinity first contacts the cluster at a boundary site is the harmonic measure of that site. Two properties of harmonic measure govern the outcome.

First, harmonic measure concentrates on convex extremities: a protruding tip subtends a large solid angle as seen from infinity, and the gradient $|\nabla u|$ there is correspondingly large. Second, harmonic measure is exponentially small deep inside a fjord, since a walker must survive many opportunities to attach on the fjord walls before reaching its base.

The consequence compounds. A tip that grows extends further into the region of high flux and captures a still larger share, while a site one branch-length behind it is screened still more severely. Growth therefore advances at the tips and stalls in the interior. As each new tip is itself an extremity subject to the same instability, the branching recurs at every scale that the cluster attains, which is the geometric content of Definition 10.1. The permanent voids visible in Figure 5 are the record: a region screened at $N=300$ receives negligible flux thereafter and stays empty at $N=9000$. $\blacksquare$

Remark 10.5 (Screening as the shape-level analogue of the barrier). The mechanism has the same logical form as the nucleation barrier, transposed from size to shape. In Theorem 4.4 a deviation in $r$ above $r_c$ was amplified by the dynamics; here a deviation of the interface outward is amplified by the flux field it itself distorts. In each case an advantage feeds back on the process that generated it, and in each case the runaway is checked only by an opposing term with different scaling: surface tension in the first case, the $\Gamma k^{3}$ term of Remark 8.4 in the second. The fractal is what the second runaway produces when it operates at all scales at once.

10.3 Mean-field estimate of the exponent

Proposition 10.6 (Witten–Sander mean-field estimate). Under the mean-field assumption that the growth velocity of the cluster envelope is proportional to the local flux of a quasi-static diffusion field in $d$ dimensions, the fractal dimension obeys
$$D ;\simeq; \frac{d^{2}+1}{d+1},$$
giving $D \simeq 5/3 \approx 1.67$ for $d = 2$.

Argument. Let $R(t)$ be the cluster radius and $M(t) \sim R^{D}$ its mass. In $d$ dimensions the quasi-static flux onto an absorbing object of size $R$ from a far-field source scales as $\dot M \sim R^{d-2}$ for $d>2$, and the growth of the envelope satisfies $\dot R \sim \dot M / (\text{active surface})$. Writing the active surface as the number of tips scaling like $R^{D-1}$, and requiring the two expressions for $\dot M$ to be mutually consistent with $M \sim R^{D}$, one obtains a single algebraic relation between $D$ and $d$ whose solution is the stated formula. The derivation treats the flux as isotropic over the envelope, which ignores the fluctuations in harmonic measure responsible for the finer structure. $\blacksquare$

Remark 10.7 (Status of the estimate). For $d=2$ the formula gives $1.67$ against the accepted numerical value $\approx 1.71$, so the mean-field argument captures the exponent to a few percent while remaining uncontrolled. The residual discrepancy has a definite origin: harmonic measure on a DLA cluster is itself multifractal, meaning a single exponent fails to describe the distribution of growth probability across the boundary, and any theory replacing that distribution by its mean loses the corresponding correction. No closed-form derivation of the exact DLA exponent is known.

This is worth stating plainly, since the surrounding sections might suggest otherwise. Sections 1–7 produced $r_{c}$ and $\Delta G^{*}$ in closed form from a two-line calculation. The analogous quantity in the fractal regime resists exact treatment, and the difference is structural: the nucleation barrier is set by a local competition at one instant, while $D$ is a property of an entire correlated growth history.

10.4 Universality and contingency

Proposition 10.8 (Universality of the exponent, contingency of the arms). Two DLA clusters grown from different random seeds have, with high probability, no branches in common, while their dimensions agree to within statistical error.

Argument. The position of any given branch traces back to a particular walker arriving at a particular site, an event of vanishing probability under a different realisation of the noise; by Proposition 9.6 that event is then frozen into the structure. The exponent, by contrast, is fixed by the screening mechanism of Proposition 10.4, which depends on the geometry of the Laplace problem and holds for every realisation. Contingent microscopic history and reproducible macroscopic exponent therefore coexist without tension. $\blacksquare$

Remark 10.9 (The two-level answer to “how does a snowflake form?”). Proposition 10.8 resolves the question that motivates this part of the note. A snowflake’s sixfold symmetry is energetic: it descends from the anisotropy of $\gamma$ in the ice lattice, is identical in every crystal, and would be predicted by an equilibrium argument. Its branch structure is historical: it descends from the particular sequence of humidity and temperature fluctuations the crystal encountered, is unique to that crystal, and is predicted by no equilibrium argument whatever. The dimension sits between the two, statistical, reproducible across crystals, and a property of the growth process, with the energy function playing no part in fixing it.

Three questions must therefore be kept apart: whether a structure forms (Theorem 3.1), what symmetry it has (energetics), and what form it takes (Sections 8–10). The apparatus answering the first answers neither of the others.

§11 Comparison of the Four Registers

Table 1. The four registers of the transition. The final row records the decisive structural difference: the first three registers track a single scalar, while the fourth requires the whole interface as its state.

	Thermodynamic	Dynamical	Stochastic	Morphological
Primary object	Landscape $\Delta G(r)$	Vector field $f(r)$	Density $P(r,t)$	Diffusion field $u$, interface
Critical object	Saddle point / maximum	Unstable fixed point	Bottleneck of flux	Unstable mode band $\omega(k)>0$
Threshold	$r_{c}=2\gamma/\Delta g_v$	Separatrix	Rate-limiting step	$k^{*}=\sqrt{v/\Gamma}$
Governing quantity	Barrier $\Delta G^{*}$	$f’(r_{c})=8\pi M\gamma$	$J\propto e^{-\Delta G^{*}/k_{\mathrm B} T}$	Dimension $D\in(1,2)$
Answers	What is paid?	What happens next?	How often?	What shape?
Cannot answer	Timescales	Origin of nuclei	Morphology	(none)
State variable	$r$ (scalar)	$r$ (scalar)	$r$ (scalar)	Full interface (field)

Remark 11.1 (The logic of the sequence). The registers stand in sequence; each supplies what the previous one lacks. Thermodynamics establishes that a barrier exists and fixes its height, but says nothing about time. Dynamics converts the landscape into a flow and shows that the barrier’s summit is a repeller separating two fates, but predicts that nothing ever happens. Stochastics supplies the fluctuations the deterministic flow discarded and recovers a finite rate whose exponent is precisely the barrier of Theorem 3.1. The chain closes: $\Delta G^{*}$ enters as a static quantity and re-emerges as the controlling exponent of a temporal one.

The fourth register is different in kind, and the table’s last row says why. The first three all track a single scalar; the fourth cannot, because morphology is irreducibly a property of the whole interface. This constitutes a change of state space, going beyond a refinement of the earlier registers, and it is the reason Proposition 9.6 can assert that no variational principle over final states selects the observed form.

§12 Concluding Propositions

Proposition 12.1 (Order arrives by threshold, never by accumulation). Since $\Delta G$ is strictly increasing on $(0,r_{c})$, no monotone accumulation of order is energetically downhill below the critical size. The system cannot reach the crystalline state by a sequence of individually favourable steps. Transition requires a fluctuation that is large enough; a process that is merely long enough in the sense of steady accumulation will fail.

Proposition 12.2 (The threshold is decisive; the endpoint is derived). By Theorem 4.4, the long-time fate of a nucleus is determined entirely by the sign of $r(0) - r_{c}$. The magnitude of the eventual crystal is not encoded in the nucleus; only the side of the separatrix is.

Proposition 12.3 (Two independent thresholds). Nucleation and morphogenesis are governed by distinct threshold conditions. The first, $r > r_{c}$, decides whether a structure persists. The second, $\omega(k) > 0$, decides what form it takes as it grows. Neither determines the other: a nucleus may cross $r_{c}$ and grow compactly, or cross it and grow fractally, according to a transport condition that plays no role in Theorem 3.1.

Proposition 12.4 (Complexity without a complex cause). The clusters of Figure 5 are produced by a rule containing no reference to branching, symmetry, or shape: particles walk at random and stick on contact. The non-integer dimension of Remark 9.5 is therefore not encoded in the rule but generated by the interaction of the rule with the geometry it progressively creates. Structural complexity in the outcome leaves the complexity of the mechanism undetermined.

Proposition 12.5 (The seed is a configuration, never a miniature). The critical nucleus contains no information about the final crystal beyond a local lattice organisation. Its causal role is exhausted by two facts: it exceeds $r_{c}$, and it presents an interface that subsequent material can extend. The macroscopic structure is produced by the environment recruited to the pattern, not stored in the seed. In the language of Theorem 4.4: the seed determines only which basin the trajectory occupies; the trajectory itself is generated by the flow.

Remark 12.6 (On transposing this structure). The apparatus above is often borrowed as an analogy for the growth of ideas, institutions, or other non-physical structures. The borrowing is legitimate only insofar as the target system genuinely exhibits the three features that do the work here:

a cost that scales with the boundary of a nascent structure and a benefit that scales with its bulk, with the boundary term dominating at small size;

a resulting non-monotone landscape, so that small deviations are actively suppressed with active suppression going beyond mere absence of reward; and

a source of undirected variation capable of producing deviations of the required size.

Sections 8–9 add a caution that the nucleation picture alone does not supply. Crossing the threshold fixes only persistence, not form (Proposition 12.3); and where growth is transport-limited and accretion irreversible, the form that results is path-dependent and admits no characterisation as an optimum (Proposition 9.6). An account that borrows the seed-and-threshold image while assuming the mature structure will be the one that some objective would have selected has taken the first half of the model and discarded the second.

Where these hold, the conclusions transfer with their sharpness intact, in particular Propositions 12.1–12.5, and the observation of Remark 3.3 that a structure poised at the threshold is the least stable of its kind. Where they do not hold, most often because (i) fails and there is no genuine interfacial penalty, the transposition yields metaphor in place of model, and it is worth being explicit about which of the two is intended.

中文

从流体到晶体

一则关于成核的热力学与动力系统笔记

黄万宏

摘要

本笔记以若干形式化的语域展开从一个亚稳流体到一个晶态固体的过渡。第 1–3 节把这一过渡当作一个自由能景观上的变分问题来处理：临界核被刻画为该景观的一个鞍点，而激活势垒被推导出来。第 4 节把同样的内容重新表述为一个一维动力系统，其中临界半径显现为一个不稳定不动点（一个排斥子），把两个吸引域分开。第 5–7 节恢复两个确定性图景都压制掉的随机成分，并显示这些语域由一个首次通过论证得以调和。第 8–9 节随后标记该装置的界限：当生长变为扩散限制时，毛细假设失效、界面变得不稳定，而由此产生的凝聚体获得一个由它自己的生长历史所固定的非整数维数，没有任何能量极小选择它。第 10 节随后推导那一区制的结构：自相似强制一个幂律、扩散场的屏蔽供出机制，而一个平均场论证估计那个指数。通篇，重点落在为什么这一过渡是阈值支配的、同时保持连续，以及落在一个小的有序构型能在何种精确意义上重组它的环境。所有图都由对正文中所陈述之方程的直接模拟产生。

关键词： 经典成核理论；临界核与自由能势垒；不稳定不动点与分界线；克拉默斯势垒穿越与首次通过时间；扩散限制凝聚与分形维数。

引言与阅读指南

关于冻结的初等说明说，在熔点之下晶体是较低能量的状态，因而流体冻结。这在一个重要方面是不完整的：它解释了目的地、而把路径留作未加说明。一个过冷液体能长时间地持续处在一个高于其自由能全局极小的状态。在那个有利的状态能被抵达之前必须付出某样东西，而那货币是界面。

整个经典成核理论都从一个不对称随之而来：体能量随体积标度，界面能量随面积标度。既然 $r^{3}$ 最终胜过 $r^{2}$、而对小的 $r$ 输给它，便有一个尺寸范围其中有序被惩罚、以及一个范围其中它被奖赏。它们之间的边界是一个真正的阈值，而这正是超出单纯描述、赋予这一现象其数学旨趣的东西。

第 1–3 节热力学地推导那个阈值。第 4 节显示它同一个流的一个不稳定平衡是同一个对象。第 5–7 节解释实际上什么使一个系统越过它。第 8–9 节问由此产生的晶体看起来像什么，并发现前述框架使这个问题完全无法回答。第 10 节供出那个缺失的说明。

§1 亚稳状态

定义 1.1（相与序参量）。 设一个单组分系统被保持在固定温度 $T$ 与压强 $p$。它的平衡态是吉布斯自由能 $G = U - TS + pV$ 的诸极小。我们以 $\varphi$ 记一个序参量：一个区分两个候选相的标量场，在无序（流体）相中 $\varphi \approx 0$，在有序（晶态）相中 $\varphi \approx 1$。物理上 $\varphi$ 可被取作平移与取向周期性的一个局部量度。

定义 1.2（亚稳性）。 一个状态 $\varphi_{0}$ 是亚稳的，如果它是 $G$ 的一个局部但非全局的极小。形式地：存在一个邻域 $\mathcal{U} \ni \varphi_{0}$ 使得对所有 $\varphi \in \mathcal{U}$ 有 $G(\varphi_{0}) \le G(\varphi)$，同时存在 $\varphi_{1} \notin \mathcal{U}$ 使得 $G(\varphi_{1}) < G(\varphi_{0})$。

注 1.3。 定义 1.2 是”液体尚未转变”这一说法的形式内容。亚稳性是一个关于 $G$ 在两个极小之间的形状的陈述：液体被这一事实所保护，即每一条通往晶体的路径都穿过更高自由能的诸状态。$-10^{\circ}\mathrm{C}$ 的过冷水对所有小扰动稳定、而只对足够大的扰动不稳定。下面的一切都是使”足够大”精确化的一个尝试。

定义 1.4（过冷度与体驱动力）。 设 $T_{m}$ 为熔化温度，并令 $\Delta T := T_{m} - T > 0$ 表示过冷度。定义体自由能密度差
$$\Delta g_v ;:=; \frac{G_{\mathrm{liquid}} - G_{\mathrm{crystal}}}{V} ;>; 0 ,$$
即每单位体积流体转化为晶体所释放的自由能。

命题 1.5（驱动力在 $T_{m}$ 附近的线性化）。 到 $\Delta T$ 的一阶，
$$\Delta g_v ;\simeq; \frac{L,\Delta T}{T_{m}} ,$$
其中 $L$ 是每单位体积的熔化潜热。

证明。 以每单位体积写 $\Delta g := g_{\ell} - g_{c} = \Delta h - T,\Delta s$。在共存处 $T = T_{m}$ 我们有 $\Delta g = 0$，因而 $\Delta s = \Delta h / T_{m} = L/T_{m}$。把 $\Delta h \simeq L$ 与 $\Delta s \simeq L/T_{m}$ 当作在小区间 $[T, T_{m}]$ 上与温度无关，
$$\Delta g(T) ;\simeq; L - T\frac{L}{T_{m}} ;=; \frac{L,(T_{m}-T)}{T_{m}} ;=; \frac{L,\Delta T}{T_{m}} . \qquad\blacksquare$$

注 1.6。 该命题记录 $\Delta g_v$ 在熔点处线性地消失。下面所推导的每一个量都在分母中依赖于 $\Delta g_v$，因而它们全都随 $T \to T_{m}^{-}$ 发散：在共存处没有驱动力、也没有成核。过冷的深度是这一理论唯一的控制参数。

§2 一个核的自由能泛函

假设 2.1（毛细近似）。 我们采纳经典成核理论的标准理想化：

（A1）锐界面。 该核是一个纯晶体区域，由一个零厚度的数学上锐利的表面同纯流体分开。

（A2）体性质。 该核的内部具有体晶体的自由能密度，无论该核多么小。

（A3）各向同性、与尺寸无关的界面张力。 该表面每单位面积耗费 $\gamma > 0$，与 $r$ 及取向无关。

（A4）球形性。 因而，由（A3）与等周不等式，固定体积下的极小化形状是一个球，由它的半径 $r \ge 0$ 参数化。

注 2.2。 这些假设对真实晶体在定量上是差的，（A3）对有刻面的材料严重失效，而（A2）对十个分子的核是可疑的，然而它们所产生的结构（一个势垒、一个临界尺寸、一个指数速率律）是稳健的，并在远为精致的处理中存续。本笔记关乎的正是那个结构。

定义 2.3（成核自由能）。 在假设 2.1 之下，在流体内部形成一个半径为 $r$ 的球形晶核的自由能耗费是函数 $\Delta G \colon [0,\infty) \to \mathbb{R}$，
$$\boxed{;\Delta G(r) ;=; \underbrace{4\pi r^{2}\gamma}{\text{界面耗费}} ;-; \underbrace{\tfrac{4}{3}\pi r^{3},\Delta g_v}{\text{体增益}} ; } \tag{1}$$

注 2.4（结构性不对称）。 方程 (1) 是两个不同次数之单项式的差，而较低的次数带正号。这单一的事实生成随之而来的一切。写 $\Delta G(r) = ar^{2} - br^{3}$ 且 $a,b>0$：当 $r \to 0^{+}$ 二次项支配，故 $\Delta G$ 增加；当 $r \to \infty$ 三次项支配，故 $\Delta G \to -\infty$。一个先上升、然后无界下降的连续函数必定拥有一个内部极大。那个势垒仅从标度指数的不匹配随之而来，无需任何附加的物理假说。

§3 临界核作为一个鞍点

定理 3.1（临界半径与激活势垒）。 设 $\Delta G$ 如 (1) 所给，$\gamma > 0$ 且 $\Delta g_v > 0$。则 $\Delta G$ 在 $(0,\infty)$ 上恰有一个内部驻点，位于
$$r_{c} ;=; \frac{2\gamma}{\Delta g_v} , \tag{2}$$
该点是一个严格局部极大，而对应的值是
$$\Delta G^{*} ;:=; \Delta G(r_{c}) ;=; \frac{16\pi\gamma^{3}}{3,\Delta g_v^{2}} ;>; 0 . \tag{3}$$
此外 $\Delta G$ 在 $(0,r_{c})$ 上严格递增、在 $(r_{c},\infty)$ 上严格递减，并在唯一的正根 $r_{0} = \tfrac{3}{2} r_{c}$ 处消失。

证明。 对 (1) 求导，
$$\Delta G’(r) ;=; 8\pi r \gamma - 4\pi r^{2}\Delta g_v ;=; 4\pi r,(2\gamma - r,\Delta g_v).$$
在 $(0,\infty)$ 上因子 $4\pi r$ 严格为正，故 $\Delta G’(r)=0$ 当且仅当 $r = r_{c} = 2\gamma/\Delta g_v$，确立唯一性。$\Delta G’$ 的符号即 $(2\gamma - r\Delta g_v)$ 的符号，它对 $r<r_{c}$ 为正、对 $r>r_{c}$ 为负；这给出单调性诸主张并显示 $r_{c}$ 是一个严格极大。作为二阶导数的确认，
$$\Delta G’’(r) = 8\pi\gamma - 8\pi r \Delta g_v, \qquad \Delta G’’(r_{c}) = 8\pi\gamma - 8\pi\frac{2\gamma}{\Delta g_v}\Delta g_v = -8\pi\gamma < 0 .$$
把 (2) 代入 (1)，
$$\Delta G^{*} = 4\pi\gamma\frac{4\gamma^{2}}{\Delta g_v^{2}} - \frac{4\pi}{3}\frac{8\gamma^{3}}{\Delta g_v^{3}}\Delta g_v = \frac{16\pi\gamma^{3}}{\Delta g_v^{2}} - \frac{32\pi\gamma^{3}}{3\Delta g_v^{2}} = \frac{16\pi\gamma^{3}}{3\Delta g_v^{2}} .$$
最后 $\Delta G(r) = 4\pi r^{2}(\gamma - \tfrac{1}{3} r \Delta g_v)$ 对 $r>0$ 消失当且仅当 $r = 3\gamma/\Delta g_v = \tfrac{3}{2}r_{c}$。$\blacksquare$

推论 3.2（随过冷度的标度）。 使用命题 1.5，
$$r_{c} ;\propto; \frac{1}{\Delta T}, \qquad \Delta G^{} ;\propto; \frac{1}{(\Delta T)^{2}} .$$
因而当 $\Delta T \to 0^{+}$ 时 $r_{c}, \Delta G^{} \to \infty$，而当熔体被进一步冷却到 $T_m$ 之下时两者都单调递减。

注 3.3（临界核作为一个过渡态）。 定理 3.1 陈述临界核沿径向坐标坐落于 $\Delta G$ 的一个极大处。在完整的构型空间中，其中该状态还有许多其他自由度，横于反应坐标 $r$，沿这些自由度 $\Delta G$ 是被极小化的，这个点是一个一阶鞍：在恰好一个方向上是极大、在所有其他方向上是极小。这正是临界核是一个过渡态的精确意义：系统穿过它而不在其上停留，而它是两个可停留状态之间耗费最小的门户。

这一后果值得强调，因为它颠倒了关于稳定性的寻常直觉。临界核是沿其自身生长方向最大程度不稳定的构型。任何更小的都衰减；任何更大的都失控；那个临界对象本身恰好被搁在分界上。阈值处的有序是最不稳固的一种有序，也是最不稳健的。

例 3.4（数量级）。 对处在 $\Delta T = 10,\mathrm{K}$ 的水，取 $\gamma \approx 30,\mathrm{mJ,m^{-2}}$ 与 $L \approx 3.0 \times 10^{8},\mathrm{J,m^{-3}}$，命题 1.5 给出 $\Delta g_v \approx 1.1 \times 10^{7},\mathrm{J,m^{-3}}$，由此
$$r_{c} \approx \frac{2(0.030)}{1.1\times 10^{7}} \approx 5.5,\mathrm{nm},$$
一个含有 $10^{4}$ 量级分子的核。对应的势垒 $\Delta G^{*}$ 是数十个 $k_{\mathrm B} T$，大到自发穿越是罕见的，而这恰恰是过冷水存在的缘由。

图 1。 (a) 定义 2.3 的景观 $\Delta G(r)$ 在三个驱动力下，标出定理 3.1 的极大。更深的过冷（更大的 $\Delta g_v$）降低并左移那个势垒，一如推论 3.2 所要求。(b) 分解为两个相竞的项。那个势垒从 $r^{2}$ 在小 $r$ 处支配 $r^{3}$、而在大 $r$ 处输给它随之而来（注 2.4）。

§4 景观上的梯度流

第 1–3 节描述了一个静态对象：一条曲线。我们现在问一个核如何在那条曲线上移动。物理上最简单可辩护的设定是，系统以正比于局部斜率的速率沿自由能下降。

假设 4.1（过阻尼梯度动力学）。 半径依照下式演化，
$$\frac{dr}{dt} ;=; -M,\frac{d,\Delta G}{dr}(r), \qquad M > 0, \tag{4}$$
其中 $M$ 是一个迁移率（动力学前因子），编码分子能多快附着到界面或从界面脱离。惯性被忽略：这是适合凝聚态物质的过阻尼极限。

定义 4.2（不动点、稳定性）。 对 $\mathbb{R}$ 上的一个流 $\dot{r} = f(r)$，一个满足 $f(r^{}) = 0$ 的点 $r^{}$ 是一个不动点（平衡）。它是线性稳定的（一个吸引子），如果 $f’(r^{*}) < 0$；是*线性不稳定的*（一个排斥子），如果 $f’(r^{*}) > 0$。

命题 4.3（显式向量场）。 在假设 2.1 与 4.1 之下，
$$\frac{dr}{dt} ;=; f(r) ;:=; -4\pi M, r,\bigl(2\gamma - r,\Delta g_v\bigr) ;=; 4\pi M \Delta g_v, r,(r - r_{c}). \tag{5}$$

证明。 由 (4) 以及定理 3.1 证明中所算出的 $\Delta G’$ 表达式，用 $r_{c} = 2\gamma/\Delta g_v$，立即得到。$\blacksquare$

定理 4.4（成核的相图）。 $[0,\infty)$ 上的流 (5) 恰有两个不动点：

$r^{}_{1} = 0$，它是稳定的*（一个吸引子），$f’(0) = -4\pi M \Delta g_v, r_{c} = -8\pi M \gamma < 0$；

$r^{}{2} = r{c}$，它是不稳定的*（一个排斥子），$f’(r_{c}) = 4\pi M \Delta g_v, r_{c} = 8\pi M \gamma > 0$。

因而 $\mathbb{R}{\ge 0}$ 分解为两个不变区间：
$$\underbrace{[0, r{c})}{0 \text{ 的吸引域}} ;;\cup;; {r{c}} ;;\cup;; \underbrace{(r_{c}, \infty)}{\text{逃逸至无界生长}} .$$
对 $r(0) \in [0,r{c})$，解满足当 $t \to \infty$ 时 $r(t) \to 0$（溶解）。对 $r(0) \in (r_{c},\infty)$，解严格递增且无界（生长）。

证明。 由 (5)，$f(r) = 4\pi M \Delta g_v, r(r-r_{c})$ 恰在 $r=0$ 与 $r=r_{c}$ 处消失。求导，$f’(r) = 4\pi M \Delta g_v,(2r - r_{c})$，在两个不动点处求值给出所陈述的符号，它们按定义 4.2 对它们分类。在 $(0,r_{c})$ 上乘积 $r(r-r_{c})$ 为负，故 $\dot{r} < 0$ 而 $r$ 单调递减；由于以 $0$ 为下界它收敛，必然收敛到一个不动点，因而收敛到 $0$。在 $(r_{c},\infty)$ 上该乘积为正，故 $r$ 单调递增；它无法收敛到一个有限极限，因为在 $r_{c}$ 之上没有不动点，因而 $r(t) \to \infty$。$\blacksquare$

图 2（相线）。 定理 4.4 的相线。实心圆：$r=0$ 处的吸引子。空心圆：$r_c$ 处的排斥子。箭头给出 $\dot r$ 的符号（在 $(0,r_c)$ 上 $\dot r < 0$，溶解／负反馈；在 $(r_c,\infty)$ 上 $\dot r > 0$，生长／正反馈），故两个吸引域携带相反的流。

推论 4.5（反馈符号）。 对一个不动点附近的小扰动写 $\delta(t) = r(t) - r^{}$。则 $\dot{\delta} \simeq f’(r^{}),\delta$，故
$$\delta(t) \simeq \delta(0),e^{f’(r^{})t}.$$
在 $r^{}{1}=0$ 处指数为负：扰动被阻尼，这是”负反馈溶解小核”的形式内容。在 $r^{*}{2}=r_{c}$ 处指数为正：扰动被放大，这是”正反馈驱动失控生长”的形式内容。两个陈述都是同一向量场在两个不同点处的一次线性化。

注 4.6（排斥子作为一条分界线）。 在一维中一个不稳定不动点恰是一条分界线：它把状态空间划分为诸吸引域，它们的未来在质上不可比较。跨 $r_{c}$ 相差一个任意小的 $\epsilon$ 的两个初始条件，有分岔的长时命运，一个消失、一个无界增长。这一敏感性完全被局域在那个阈值处；离开它，动力学是乏味且可预测的。

请注意同一个点在两个语域中带两个不同的名字。沿反应坐标，$\Delta G$ 的极大（定理 3.1）就是那个流的排斥子（定理 4.4）。两者由一个恒等式相关：对一个梯度流 $\dot r = -M,\Delta G’(r)$ 有 $f’(r) = -M,\Delta G’’(r)$，故一个不动点的稳定性是该景观的符号翻转的曲率。极大排斥、极小吸引。两个形式体系是同一陈述在两个方向上被读。

注 4.7（关于分岔理论的读法）。 把成核描述为一个亚临界分岔是常见的，而这个类比是有用的，但一则告诫是适宜的。严格地，一个分岔是不动点的数目或稳定性随一个控制参数被改变而发生的一个改变。此处自然的控制参数是过冷度 $\Delta T$。由 (5) 与推论 3.2，当 $\Delta T \to 0^{+}$ 我们有 $r_{c} \to \infty$：排斥子退到无穷、而流体变得全局稳定。当 $\Delta T$ 增加，$r_{c}$ 下降到分子尺度，而一旦 $r_{c}$ 趋近一个单一结构单元的尺寸，那个势垒便不再有意义，系统对任何涨落都变得不稳定。那个极限是旋节线，而正是在那里、而非在有限过冷处，一个真正的局部稳定性丧失发生。

那么对固定的 $\Delta T$，我们所展示的是一个带一个阈值的双稳结构：由一个排斥子分开的两个吸引域，不涉及任何分岔。这一区分要紧。一个分岔改变什么状态存在；一个阈值支配诸既存状态中哪一个被抵达。

图 3。 (a) 命题 4.3 的向量场 $\dot r=-M,\Delta G’(r)$。阴影标出 $\dot r$ 的符号；实心圆是 $r=0$ 处的吸引子，空心圆是 $r_c$ 处的排斥子。(b) 同一个流的诸轨迹。在 $r_c$ 附近相差 $\pm 0.03$ 的初始条件分离到相反的命运，这是注 4.6 的分界线性质。

§5 确定性描述的界限

注 5.1（那个悖论）。 定理 4.4 含有一个尴尬。流体状态 $r=0$ 是一个稳定吸引子：$r_{c}$ 之下的每一个初始条件都返回到它。照字面理解，确定性的流 (5) 预测一个过冷液体永不冻结，因为它从 $r=0$ 开始并停留在那里。

那个预测是假的，而它的假性恰好定位了那个缺失的成分。方程 (4) 描述半径的平均漂移，却丢弃了核起初由之生起的热搅动。确定性动力学能解释一个核一旦存在会发生什么；它无法解释一个核的存在。有序从围绕系统之系统性倾向的诸涨落中生起；那个倾向本身指向另一个方向。

§6 朗之万与福克-普朗克表述

定义 6.1（核半径的朗之万方程）。 恢复那个热项：
$$\frac{dr}{dt} ;=; -M,\frac{d,\Delta G}{dr}(r) ;+; \eta(t), \tag{6}$$
其中 $\eta$ 是高斯白噪声，满足
$$\langle \eta(t)\rangle = 0, \qquad \langle \eta(t)\eta(t’)\rangle = 2D,\delta(t-t’),$$
而扩散系数 $D$ 由涨落-耗散关系 $D = M k_{\mathrm B} T$ 同迁移率绑定。

注 6.2。 关系 $D = M k_{\mathrm B} T$ 携带超出建模便利的物理内容。它陈述那同一些阻碍生长（耗散，$M$）的分子碰撞，正是生成涨落（$D$）的那些。人不可能有一个抵抗运动却不抖动的系统；一个有能力摧毁一个核的介质由此有能力创造一个。摧毁的机制与创造的机制是同一个机制。

定义 6.3（福克-普朗克方程）。 设 $P(r,t)$ 为在时间 $t$ 找到一个半径为 $r$ 之核的概率密度。对应于 (6)，$P$ 服从
$$\frac{\partial P}{\partial t} ;=; \frac{\partial}{\partial r}!\left[ M,\frac{d,\Delta G}{dr},P ;+; D,\frac{\partial P}{\partial r} \right] ;=; -\frac{\partial J}{\partial r}, \tag{7}$$
其中
$$J(r,t) ;=; -M\frac{d,\Delta G}{dr}P - D\frac{\partial P}{\partial r} \tag{8}$$
是尺寸空间中的概率流。

注 6.4（读方程 (7)）。 (7) 中的方括号在相邻的诸项中含有整个理论的两个相竞倾向。第一个是漂移：沿景观确定性地滑下，它对 $r<r_{c}$ 指向溶解。第二个是扩散：不顾能量学的概率铺展，它把一些密度推向上坡。成核是这一竞争的残余，即扩散越过一个漂移正试图强制执行之势垒所带过的那一小份概率泄漏。

命题 6.5（平衡分布）。 (7) 在零流 $J \equiv 0$ 下的定态解是玻尔兹曼分布
$$P_{\mathrm{eq}}(r) ;=; \mathcal{N}\exp!\left(-\frac{\Delta G(r)}{k_{\mathrm B} T}\right).$$

证明。 在 (8) 中令 $J=0$ 给出 $D,P’ = -M,\Delta G’,P$，即用 $D = M k_{\mathrm B} T$ 有 $(\ln P)’ = -(M/D),\Delta G’ = -\Delta G’/k_{\mathrm B} T$。积分得到所陈述的形式。$\blacksquare$

注 6.6。 命题 6.5 已然含有那个本质的指数。它说亚临界核保持罕见却总是在场，以权重 $e^{-\Delta G(r)/k_{\mathrm B} T}$ 被布居。一个过冷液体在任何时刻都含有一群小的瞬态有序团簇，其中几乎全部都溶解。成核是持续地被创造与被摧毁之有序的偶尔存活。

§7 成核速率作为一个首次通过问题

定义 7.1（成核速率）。 定态成核速率 $J_{\mathrm{nuc}}$ 是 (8) 中的定态概率流，处在这样的边界条件之下：在小 $r$ 处有一个源（维持亚临界团簇的近平衡布居），在大 $r$ 处有一个吸收汇（超临界核被移除到宏观生长）。等价地，它是从 $r \approx 0$ 跨过 $r_{c}$ 的平均首次通过时间的倒数。

定理 7.2（克拉默斯／阿伦尼乌斯形式）。 对 $\Delta G^{} \gg k_{\mathrm B} T$，定态速率服从
$$\boxed{; J_{\mathrm{nuc}} ;=; A \exp!\left(-\frac{\Delta G^{}}{k_{\mathrm B} T}\right) ;=; A \exp!\left(-\frac{16\pi\gamma^{3}}{3,\Delta g_v^{2},k_{\mathrm B} T}\right) ; } \tag{9}$$
带一个由迁移率、以及由 $\Delta G$ 在两个相关点处的曲率所决定的动力学前因子 $A$。

证明概要。 在 (8) 中施加一个常流 $J$，并以积分因子 $e^{\Delta G(r)/k_{\mathrm B} T}$ 求解由此得到的一阶线性常微分方程，给出
$$J \int_{0}^{R} \frac{e^{\Delta G(r)/k_{\mathrm B} T}}{D},dr ;=; P(0)e^{\Delta G(0)/k_{\mathrm B} T} - P(R)e^{\Delta G(R)/k_{\mathrm B} T}.$$
在一个吸收条件 $P(R)\to 0$ 以及 $P(0)$ 由局部平衡设定之下，左边的积分对 $\Delta G^{}\gg k_{\mathrm B} T$ 由极大 $r_{c}$ 的邻域支配。那里的拉普拉斯方法，把 $\Delta G(r) \simeq \Delta G^{} + \tfrac12 \Delta G’’(r_{c})(r-r_{c})^{2}$ 展开、其中 $\Delta G’’(r_{c}) = -8\pi\gamma < 0$，产生因子 $e^{\Delta G^{*}/k_{\mathrm B} T}$ 乘以一个高斯宽度，而取倒数给出 (9)。$\blacksquare$

推论 7.3（对过冷度的极端敏感性）。 把 (9) 同推论 3.2 结合，
$$J_{\mathrm{nuc}} ;\propto; \exp!\left(-\frac{C}{T,(\Delta T)^{2}}\right)$$
其中 $C$ 是一个材料常数。因而该速率以双重指数的锐利度随 $\Delta T$ 增加：在一个狭窄的过冷窗口内，$J_{\mathrm{nuc}}$ 能改变许多个数量级。这正是为什么冻结显得有一个明确定义的起始温度，尽管它形式上是 $T$ 的一个连续函数。

图 4。 (a) 从 $r=0.05$ 起始的确定性流：它返回到 $0$ 并永远停留在那里，即注 5.1 的悖论。(b) 从同一初始条件出发的十二条朗之万轨迹（定义 6.1）；那些抵达吸收边界的以红色绘出。噪声、而非漂移，产生了那个过渡。(c) 命题 6.5 的玻尔兹曼布居：亚临界团簇是指数地罕见、却从不缺席。(d) 每个温度 $120$ 条轨迹得出的穿越速率对 $1/k_BT$。斜率估计出一个有效势垒 $13.5$，对照理论值 $\Delta G^{}=16.8$；这个差距是预期之中的，因为定理 7.2 在 $\Delta G^{}/k_BT$ 中是渐近的、而前因子 $A$ 本身弱依赖于温度。

§8 毛细模型的范围

注 8.1（该模型的一个边界）。 迄今所确立的一切都关乎一个单一标量，即半径 $r$，并回答一个单一问题：该核是否存活？ 假设 2.1 以一个如今必须偿付的代价购得这一简单性。条件（A3）与（A4），即各向同性的 $\gamma$、球形，超出了简化几何：它们预设了关于形式之问题的答案。一个单参数模型无法产生一个形状，因为它已然假定了一个。

因而从第 1–7 节到随之而来者的过渡是一次区制的改变，有别于同一模型延续到一个更晚的时刻。下面的分形形貌恰恰在（A3）–（A4）失效之处生起，而那个分析任务是辨认什么使它们失效。

定义 8.2（两个速率限制区制）。 设 $\tau_{D}$ 为一个分子扩散到生长前沿的特征时间，$\tau_{A}$ 为它一旦到达便附着的特征时间。定义诸区制：

（R1）附着限制（$\tau_{A} \gg \tau_{D}$）：输运是快的，晶体周围的浓度场近乎均匀，而一个未能在一个位点附着的分子采样许多其他位点。生长由界面能量学支配。

（R2）扩散限制（$\tau_{D} \gg \tau_{A}$）：输运是慢的，分子基本上在它首先到达之处附着，而浓度场在诸凸起周围发展出陡峭的梯度。

命题 8.3（区制（R2）打破那个毛细假设）。 在区制（R2）中，一个与尺寸无关、形状中性的界面惩罚（A3）这一假设不再控制形貌。控制转到扩散场，它是整个既存形状的一个非局部泛函。

论证。 设 $u$ 为生长单元的浓度，满足准静态扩散问题 $\nabla^{2}u = 0$（在晶体之外），$u$ 在界面处与无穷远处被固定。局部生长速度是 $v_{n} \propto \partial u/\partial n$。对一个调和函数，法向梯度在边界最凸之处最大：诸凸起截获来临通量中不成比例的一份，而诸凹处被屏蔽。因而一个高度为 $h$ 的凸起以一个随 $h$ 递增的速率生长。

这是一个对形状的正反馈，在结构上平行于定理 4.4 中对尺寸的正反馈，并有同样的后果：超出一个阈值的扰动经历放大。界面张力对抗它，因为曲率通过吉布斯-汤姆孙效应抬高局部平衡浓度，而这恰恰惩罚最尖锐的特征。因而这一竞争选择一个有限波长，排除光滑的球与无限精细的枝晶两者。当扩散驱动力大时，不稳定波长的带宽是宽的、被选择的特征是精细的，而（A4）在每一个尺度上同时被违反。$\blacksquare$

注 8.4（马林斯-塞克尔卡不稳定性）。 命题 8.3 是一个经典线性稳定性结果的定性陈述。以一个波数为 $k$、振幅为 $\delta_{k}$ 的模式扰动一个生长的界面，示意地给出
$$\frac{1}{\delta_{k}}\frac{d\delta_{k}}{dt} ;=; \omega(k) ;\sim; \underbrace{v,k}{\text{扩散失稳}} ;-; \underbrace{\Gamma k^{3}}{\text{毛细稳定}},$$
其中 $\Gamma$ 正比于 $\gamma$。这两个项携带 $k$ 的不同幂次，即注 2.4 的同一结构性手法，如今在波数空间中以 $k$ 代 $r$。短波长被表面张力稳定（$k^{3}$ 支配），长波长被扩散场失稳（$vk$ 支配），而 $\omega(k)$ 在 $k^{*} = \sqrt{v/\Gamma}$ 处变号。图 6(e) 为两个驱动力绘出 $\omega$。

请注意这一模式的复现：在两种情形中，一个带较高幂次的稳定机制在一个极限中输给一个带较低幂次的失稳机制、而在另一个极限中胜出，而那个交叉在一个阈值处发生，两者之间没有渐进的权衡。

§9 扩散限制凝聚与分形维数

定义 9.1（DLA）。 扩散限制凝聚是区制（R2）的极简随机模型。一个种子粒子被固定在原点。诸粒子每次一个地从一个遥远的边界被释放、执行随机游走，并在接触该团簇时永久地成为它的一部分。可施加一个粘附概率 $s \in (0,1]$：接触时游走者以概率 $s$ 附着、否则继续游走。

注 9.2（为什么 $s$ 在两个区制之间插值）。 参数 $s$ 是该模型对定义 8.2 的控制。当 $s = 1$ 时游走者在首次接触时附着，必然在外包络上：这是区制（R2）。当 $s \ll 1$ 时游走者拒绝大多数接触并继续扩散，故它穿入诸枝之间的峡湾并填充它们：形状无关性被恢复，而该模型趋近区制（R1）。因而一个参数横跨两个区制，而其形貌后果可被直接读出。

定义 9.3（盒计数维数）。 对一个有界集 $S \subset \mathbb{R}^{2}$，设 $N(\epsilon)$ 为一个覆盖网格中与 $S$ 相交、边长为 $\epsilon$ 的盒子的数目。盒计数维数是
$$D ;=; \lim_{\epsilon \to 0} \frac{\log N(\epsilon)}{\log(1/\epsilon)},$$
在实践中估计为 $\log N$ 对 $\log(1/\epsilon)$ 在一个尺度范围上的斜率。

定义 9.4（质量-半径指数）。 设 $M(R)$ 为距种子 $R$ 距离之内的粒子数。若 $M(R) \sim R^{D_{M}}$ 在一个 $R$ 范围上成立，则指数 $D_{M}$ 是质量维数。对一个紧致的平面对象 $D_{M} = 2$；一个 $D_{M} < 2$ 的值指示该结构在任何尺度都未能填满平面。

注 9.5（数值结果）。 图 5 显示一个 $9000$ 粒子的扩散限制团簇的生长历史，颜色编码附着次序。图 6 比较两个区制。测得的值是

盒计数 $D$ 质量指数 $D_{M}$

扩散限制 $(s=1)$ 1.54 1.56

附着限制 $(s=0.03)$ 1.73 1.83

两点观察。第一，两个独立的估计量在每一区制之内一致，这充当对测量的一个一致性检查，供出无附加结果。第二，且实质地，扩散限制团簇的 $D$ 严格介于 $1$ 与 $2$ 之间：它既非一条曲线、亦非一个填满的区域。附着限制团簇明显更稠密，并趋向空间填充值 $2$。

为对精度诚实：二维 DLA 公认的渐近值是 $D \approx 1.71$，而此处所得的值 $1.54$ 被有限尺寸效应压低，因为在 $9000$ 粒子时那个标度窗口跨越不足两个数量级。该图证明一个非整数维数的存在以及那个区制差别的方向；它的精度达不到对 DLA 指数的一次精确测量。

	盒计数 $D$	质量指数 $D_{M}$
扩散限制 $(s=1)$	1.54	1.56
附着限制 $(s=0.03)$	1.73	1.83

图 5。 从一个单一种子（$s=1$）的扩散限制生长，处于四个阶段。颜色编码附着次序（深 = 早）。分枝形式在 $N \approx 300$ 时便在场，并随团簇扩大而被保留、而非被生长掉：更晚的粒子延伸既存的尖端、而使内部保持未填充。没有分枝规则被施加；那个形貌从命题 8.3 的屏蔽论证随之而来。

图 6。 定义 8.2 的两个区制。(a) 扩散限制（$s=1$）：分叉的，带更晚粒子永不抵达的深峡湾。(b) 附着限制（$s=0.03$）：同一算法带拒绝，产生一个紧致团簇，即假设 2.1 近似有效的那个区制。(c) $D$ 的盒计数估计。(d) 质量-半径标度，紧致参照 $M \sim R^{2}$ 以虚线示出。(e) 注 8.4 的稳定性谱 $\omega(k)$ 在两个驱动力下；不稳定模式 $\omega>0$ 的带宽随驱动力变宽。

命题 9.6（分形性作为被冻结的历史）。 在区制（R2）中，那个形貌未能极小化任何自由能泛函。它记录材料到达的次序。

论证。 DLA 中的附着是不可逆的：一个加入团簇的粒子从不脱离，故系统不采样诸构型、无法向一个极小弛豫。命题 8.3 的屏蔽随后意味每一次附着改变所有后续附着的通量场，故早期事件不对称地约束晚期事件。图 5 直接展现这一点：深色（早期）的核通篇保持它的形状，而在 $N=300$ 时可见的内部空洞在 $N=9000$ 时仍是空洞，因为它们从它们形成的那一刻起便被屏蔽。因而最终对象是一个路径依赖的结果，而没有任何关于最终状态的变分原理能选择它。$\blacksquare$

注 9.7（对平衡图景的后果）。 命题 9.6 标记热力学语域的真正界限。第 1–3 节能计算一个临界尺寸，因为那个量由单一时刻的一个能量竞争决定，不涉及历史。区制（R2）中的形貌没有这样的刻画。这正是为什么雪花在个体上各异、却全都共享六重对称：那个对称是能量的、因而是普适的，而那个分枝结构是历史的、因而是特定于穿过一个涨落环境的一条轨迹的。

§10 分形凝聚体形成的理论

第 9 节测量了一个维数。本节推导为什么一个那种维数根本存在、为什么它严格介于 $1$ 与 $2$ 之间，以及为什么它是可复现的、而那些个体分枝不是。

10.1 自相似与非整数维数

定义 10.1（统计自相似）。 一个随机集 $S$ 在一个尺度范围 $[\ell_{\min}, \ell_{\max}]$ 上是统计自相似的，如果对该范围中的 $\lambda$，被重标度的集 $\lambda S$ 与 $S$ 有同样的统计分布，直到一个总体尺寸的改变。这个要求是关于系综的：个体实现各异，而它们的统计是标度不变的。

命题 10.2（自相似强制一个幂律）。 设 $M(R)$ 为距种子半径 $R$ 之内的质量。若该凝聚体在定义 10.1 的意义上是统计自相似的，则 $M$ 服从一个幂律 $M(R) = c R^{D}$，其中 $D$ 是某个指数，且 $D$ 是定义 9.4 的质量维数。

证明。 自相似意味期望质量剖面对所有可容许的 $\lambda, R$ 满足 $M(\lambda R) = g(\lambda)M(R)$，其中 $g$ 仅依赖于 $\lambda$。固定 $R_{0}$ 并令 $\phi(u) = M(R_{0}e^{u})/M(R_{0})$。该关系变为 $\phi(u+v) = \phi(u)\phi(v)$，即柯西指数方程。在 $\phi$ 可测且为正的情况下，它的解是 $\phi(u) = e^{Du}$，其中 $D$ 为一个常数。代回，$M(R) = M(R_{0})(R/R_{0})^{D}$。$\blacksquare$

注 10.3（那个指数意味什么）。 命题 10.2 解释为什么图 6(d) 的对数-对数图是直的：那里的一条直线就是标度不变性的陈述，而它的斜率是剩下的唯一自由参数。两个整数情形夹住诸可能性。一条光滑曲线给出 $D=1$；一个填满的区域给出 $D=2$。一个严格介于它们之间的值描述一个随生长变得稀疏的对象：既然半径 $R$ 之内的平均密度标度为
$$\rho(R) ;=; \frac{M(R)}{\pi R^{2}} ;\propto; R^{D-2} ;\xrightarrow[R \to \infty]{} 0 \qquad \text{每当 } D < 2,$$
一个无限 DLA 团簇在平面中有消失的密度。该凝聚体占据比一条曲线更多的空间，同时填满它可用面积的一个消失的份额，而指数 $D$ 恰好量化它落在那两个极限之间的何处。

10.2 屏蔽机制

命题 10.4（屏蔽产生标度不变的稀疏性）。 在区制（R2）中，生长概率在生长的每一阶段都集中在团簇的外部末端。因而内部保持永久地欠填充，而这个亏缺在每一尺度上被复现。

论证。 设 $u$ 解命题 8.3 的外部拉普拉斯问题，团簇被保持在固定势、无穷远处有一个源。一个来自无穷远的随机游走者在一个边界位点首次接触团簇的概率是那个位点的调和测度。调和测度的两个性质支配这一结果。

第一，调和测度集中在凸的末端：一个凸出的尖端从无穷远看去张开一个大的立体角，而那里的梯度 $|\nabla u|$ 相应地大。第二，调和测度在一个峡湾的深处是指数地小，因为一个游走者必须在抵达其底部之前挺过许多次在峡湾壁上附着的机会。

这一后果复利。一个生长的尖端更远地延伸进高通量的区域并捕获一个还要更大的份额，而一个在它后方一个枝长处的位点被还要更严重地屏蔽。因而生长在尖端处推进、在内部处停滞。由于每一个新尖端本身是一个受制于同一不稳定性的末端，那个分枝在团簇所达到的每一尺度上复现，这正是定义 10.1 的几何内容。图 5 中可见的永久空洞是那个记录：一个在 $N=300$ 时被屏蔽的区域此后接收可忽略的通量，并在 $N=9000$ 时保持空。$\blacksquare$

注 10.5（屏蔽作为势垒在形状层面的类似物）。 该机制与成核势垒有同样的逻辑形式，从尺寸转置到形状。在定理 4.4 中一个 $r$ 在 $r_c$ 之上的偏差被那个动力学放大；此处一个界面向外的偏差被它自身所扭曲的通量场放大。在每一情形中一个优势反馈到产生它的那个过程之上，而在每一情形中那个失控只被一个带不同标度的相反项所制衡：第一种情形中是表面张力，第二种情形中是注 8.4 的 $\Gamma k^{3}$ 项。那个分形是当第二个失控在所有尺度上同时运作时所产生之物。

10.3 那个指数的平均场估计

命题 10.6（维滕-桑德平均场估计）。 在这样的平均场假设之下，即团簇包络的生长速度正比于 $d$ 维中一个准静态扩散场的局部通量，那个分形维数服从
$$D ;\simeq; \frac{d^{2}+1}{d+1},$$
对 $d = 2$ 给出 $D \simeq 5/3 \approx 1.67$。

论证。 设 $R(t)$ 为团簇半径、$M(t) \sim R^{D}$ 为其质量。在 $d$ 维中，从一个远场源到一个尺寸为 $R$ 之吸收对象上的准静态通量对 $d>2$ 标度为 $\dot M \sim R^{d-2}$，而包络的生长满足 $\dot R \sim \dot M / (\text{活跃表面})$。把活跃表面写作标度像 $R^{D-1}$ 的尖端数目，并要求 $\dot M$ 的两个表达式与 $M \sim R^{D}$ 相互一致，便得到 $D$ 与 $d$ 之间的一个单一代数关系，其解是所陈述的公式。这一推导把通量当作在包络上各向同性，这忽略了对更精细结构负责的调和测度的诸涨落。$\blacksquare$

注 10.7（该估计的地位）。 对 $d=2$ 那个公式给出 $1.67$，对照公认的数值值 $\approx 1.71$，故那个平均场论证把那个指数捕捉到百分之几，同时仍不受控。那个残余差异有一个确定的来源：一个 DLA 团簇上的调和测度本身是多重分形的，意味一个单一指数未能描述生长概率跨边界的分布，而任何以它的均值替换那个分布的理论都丢失对应的修正。DLA 精确指数的闭形式推导尚无人知晓。

这值得直白陈述，因为周围诸节可能暗示相反。第 1–7 节从一个两行的计算以闭形式产生了 $r_{c}$ 与 $\Delta G^{*}$。分形区制中的类似量抗拒精确处理，而那一差别是结构性的：成核势垒由单一时刻的一个局部竞争设定，而 $D$ 是整个相关生长历史的一个性质。

10.4 普适性与偶然性

命题 10.8（指数的普适性，臂的偶然性）。 从不同随机种子生长的两个 DLA 团簇，高概率地没有共同的分枝，而它们的维数在统计误差之内一致。

论证。 任何给定分枝的位置追溯到一个特定游走者在一个特定位点的到达，一个在噪声的一个不同实现之下概率消失的事件；由命题 9.6 那个事件随后被冻结进那个结构。相比之下，那个指数由命题 10.4 的屏蔽机制所固定，它依赖于拉普拉斯问题的几何、并对每一个实现成立。偶然的微观历史与可复现的宏观指数因而毫无张力地共存。$\blacksquare$

注 10.9（对”一片雪花如何形成？”的两层回答）。 命题 10.8 解决激发本笔记这一部分的那个问题。一片雪花的六重对称是能量的：它从冰晶格中 $\gamma$ 的各向异性下传、在每一个晶体中都相同，并会被一个平衡论证所预测。它的分枝结构是历史的：它从该晶体所遭遇的湿度与温度涨落的特定序列下传、对那个晶体是独一无二的，且不被任何平衡论证所预测。那个维数坐落于两者之间，即统计的、跨诸晶体可复现的，并是那个生长过程的一个性质，那个能量函数在固定它上不起任何作用。

因而三个问题必须被分开：一个结构是否形成（定理 3.1）、它有什么对称（能量学），以及它随生长采取什么形式（第 8–10 节）。回答第一个的装置回答不了另外两个中的任何一个。

§11 四个语域的比较

表 1。 这一过渡的四个语域。最后一行记录那个决定性的结构差别：前三个语域追踪一个单一标量，而第四个要求整个界面作为它的状态。

	热力学	动力学	随机	形貌
主要对象	景观 $\Delta G(r)$	向量场 $f(r)$	密度 $P(r,t)$	扩散场 $u$、界面
临界对象	鞍点／极大	不稳定不动点	通量的瓶颈	不稳定模式带 $\omega(k)>0$
阈值	$r_{c}=2\gamma/\Delta g_v$	分界线	速率限制步骤	$k^{*}=\sqrt{v/\Gamma}$
支配量	势垒 $\Delta G^{*}$	$f’(r_{c})=8\pi M\gamma$	$J\propto e^{-\Delta G^{*}/k_{\mathrm B} T}$	维数 $D\in(1,2)$
回答	付出什么？	接下来发生什么？	多频繁？	什么形状？
无法回答	时间尺度	核的起源	形貌	（无）
状态变量	$r$（标量）	$r$（标量）	$r$（标量）	整个界面（场）

注 11.1（这一序列的逻辑）。 诸语域处在序列之中；每一个供出前一个所缺之物。热力学确立一个势垒存在并固定它的高度，却对时间什么也没说。动力学把景观转换为一个流，并显示那个势垒的顶峰是一个把两个命运分开的排斥子，却预测无事发生。随机学供出确定性流所丢弃的诸涨落，并恢复一个有限速率，其指数恰恰是定理 3.1 的那个势垒。这条链闭合：$\Delta G^{*}$ 作为一个静态量进入，并作为一个时间性量的控制指数重新浮现。

第四个语域在种类上不同，而该表的最后一行说明缘由。前三个都追踪一个单一标量；第四个无法，因为形貌不可化约地是整个界面的一个性质。这构成状态空间的一次改变，超出对早先诸语域的一次精化，而它是命题 9.6 能断言没有任何关于最终状态的变分原理选择被观察到之形式的缘由。

§12 结论性诸命题

命题 12.1（有序按阈值到来，从不按累积）。 既然 $\Delta G$ 在 $(0,r_{c})$ 上严格递增，在临界尺寸之下没有任何有序的单调累积在能量上是下坡的。系统无法通过一个个个体上有利的步骤序列抵达晶态。过渡要求一个足够大的涨落；一个仅仅在稳定累积意义上足够久的过程将失败。

命题 12.2（阈值是决定性的；终点是被导出的）。 由定理 4.4，一个核的长时命运完全由 $r(0) - r_{c}$ 的符号所决定。最终晶体的大小不被编码在那个核之中；只有分界线的哪一侧被编码。

命题 12.3（两个独立的阈值）。 成核与形态发生由不同的阈值条件支配。第一个，$r > r_{c}$，决定一个结构是否持续。第二个，$\omega(k) > 0$，决定它随生长采取什么形式。二者都不决定对方：一个核可能跨过 $r_{c}$ 并紧致地生长，或跨过它并分形地生长，取决于一个在定理 3.1 中不起任何作用的输运条件。

命题 12.4（没有一个复杂原因的复杂性）。 图 5 的诸团簇由一个不含对分枝、对称或形状之任何指涉的规则所产生：诸粒子随机游走并在接触时粘附。因而注 9.5 的那个非整数维数不被编码在那个规则之中、而由那个规则与它逐步创造的几何之间的互动所生成。结果中的结构复杂性使那个机制的复杂性未被决定。

命题 12.5（那个种子是一个构型，从不是一个缩微物）。 临界核除了一个局部的晶格组织之外不含关于最终晶体的任何信息。它的因果角色被两个事实穷尽：它超过 $r_{c}$，且它呈现一个后续材料能延伸的界面。宏观结构由被招募到那个模式的环境所产生，而非被储存在那个种子之中。用定理 4.4 的语言：那个种子只决定那条轨迹占据哪个吸引域；那条轨迹本身由那个流所生成。

注 12.6（关于转置这一结构）。 上面的装置常被借作对观念、制度或其他非物理结构之生长的一个类比。这一借用只在目标系统真正展现在此处起作用的三个特征时才是正当的：

一个随一个初生结构的边界标度的耗费，以及一个随它的体标度的收益，而边界项在小尺寸处支配；

一个由此产生的非单调景观，从而小偏差被主动地压制，而主动压制超出单纯的奖赏之缺席；以及

一个有能力产生所需尺寸之偏差的无定向变异来源。

第 8–9 节添加一则单靠成核图景不供出的告诫。跨过那个阈值只固定持续、而非形式（命题 12.3）；而在生长是输运限制的、附着是不可逆的之处，由此产生的形式是路径依赖的，且不容许被刻画为一个最优（命题 9.6）。一个借用种子-与-阈值意象、同时假定成熟结构将是某个目标本会选择之物的说明，取了这个模型的前一半而丢弃了后一半。

在这些成立之处，那些结论以其锐利度完好地转移，尤其是命题 12.1–12.5，以及注 3.3 的观察，即一个被搁在阈值处的结构是其同类中最不稳定的。在它们不成立之处，最常因为（i）失效、没有真正的界面惩罚，那个转置产出隐喻以代模型，而就意图的是两者中的哪一个保持明确是值得的。