A Multi-Scale Generative Grammar for Crystal Formation - Lattice, Morphology, and the Coupling Between Them 【（Preliminary）Draft】

Posted on 2026-07-26 In Physics , Statistical Mechanics Views: Disqus: Word count in article: 16k Reading time ≈ 57 mins.

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A Multi-Scale Generative Grammar for Crystal Formation

Lattice, Morphology, and the Coupling Between Them

Wanhong Huang · Independent Researcher · huangwanhong@serendip.ngo

Abstract

Formal models of physical formation are usually models of states: a configuration is assigned an energy, and the theory selects among configurations. Processes that are irreversible and history-dependent resist this treatment, since their outcomes are distinguished by how they were produced and by nothing about the product itself. Crystal formation contains both kinds at once. Its lattice is selected by energy and is identical in every specimen; its branched morphology is selected by the order of accretion and is unique to one history. This paper models both in a single formalism, generative grammar, and locates the difference between them inside that formalism.

Three observations are offered. First, a dynamical process can be written as a generative grammar in which the derivation carries the physics, which makes procedural content a formal object and brings the associated theorems within reach. Second, the two scales of crystal formation appear to occupy different components of a grammar: the lattice scale is a statement about the language, order-independent and energetically selected, while the morphological scale is a statement about the derivation, path-dependent and admitting no variational characterisation; weights on configurations alone then look insufficient to reproduce the statistics of growth. Third, the two scales couple at a single point of the formalism, the applicability condition of one production: the lattice alphabet restricts where growth may occur. Small computations reported here agree with the published simulation literature that this restriction fixes the arm count and orientation of the cluster, and that literature reports it shifting the fractal dimension from the continuum value toward a rigorous bound.

Each of the three models is given explicitly, with its production rules, a worked example, and a figure. The lattice-scale formalism is due to others; what is attempted here is a placement of the parts.

Keywords: generative grammar; crystal growth; substitution rules; diffusion-limited aggregation; fractal dimension.

Note. This is a draft and discussion paper. Objections, discussion, and corrections are welcome at the address above.

Introduction

A crystal has a repeating unit and a shape. The repeating unit is fixed by energetics and is shared across every specimen of the substance. The shape is not: two snowflakes of identical lattice symmetry agree in no branch.

The standard apparatus of statistical physics addresses the first of these and is silent about the second, for a structural reason. That apparatus assigns a weight to a configuration and sums over configurations, so everything it computes is a function of the product. Where a process is reversible and equilibrating this loses nothing, since the system samples its configurations and the history is washed out. Where accretion is irreversible and the arriving material never detaches, the system samples nothing, and two objects that are indistinguishable as configurations may have been overwhelmingly unequal in the probability of their formation. A theory that indexes only configurations cannot state this difference, and the classical treatment of nucleation, which computes a critical size and an activation barrier by a local competition of energies at one instant, has no resources for it either.

What is required is a formalism in which the procedure is a first-class object, carrying its own structure and its own theorems, alongside the product it generates.

Generative grammar separates exactly the components that must be distinguished. A grammar consists of a production set, a language, and a derivation. The language is the set of objects the grammar admits and is a property of the grammar. A derivation is one route through the productions and is a property of a single history. Two derivations may terminate in the same string.

Four features make this more than a change of notation. The productions are a finite, inspectable list, so a constraint on the process becomes a syntactic object that can be edited, compared, and reasoned about. The distinction between language and derivation is already load-bearing in the formalism, so a physical claim can be assigned to one or the other and the assignment carries consequences. The Chomsky hierarchy supplies a measure of the resources a process demands, which is a meaningful classification of the process itself. And composition of grammars is well understood, which is what a multi-scale account requires: the relation between a description at one length scale and a description at another has a counterpart in the morphisms between grammars, with block-spin renormalisation appearing among them.

The formalism has already been applied to periodic and aperiodic order, where it is rigorous and categorical. That work treats equilibrium structures, and the objects it assigns to a physical system are languages. The procedural side has received no comparable treatment, and it is the side on which the questions of this paper fall.

The paper offers four things, none of them large.

A grammatical presentation of a growth process (§3). Diffusion-limited growth is written as a grammar with a single production whose applicability is governed by harmonic measure. In this presentation the production set is nearly trivial and the derivation carries the content, which restates in grammatical terms the familiar observation that such a cluster records its own history. Two small propositions accompany it: the selection rule is nonlocal, which places the grammar above context-free without fixing its level; and the probability of a configuration depends on more than the configuration, so weights on configurations alone appear insufficient to reproduce the statistics.
An asymmetry between the two scales (§4). The suggestion is that the two scales of crystal formation place their physical content in different components of the grammar, the language at the lattice scale and the derivation order at the morphological scale. The observation is consistent with the familiar split in what is reproducible across specimens, and it offers a vocabulary for that split; it is not established that the vocabulary buys anything beyond description.
A coupling located at one production (§5). The lattice alphabet can be read as a restriction on where the growth production may fire. Small computations reported here are consistent with that restriction fixing the arm count and orientation of the cluster, in agreement with the published simulation literature; the further claim, that the restriction shifts the fractal dimension toward the bound proved by Kesten, is taken from that literature and is beyond the reach of anything computed here.
Worked examples. Each of the three grammars is given with its productions, a small example computed in full, and a figure. The examples are chosen for transparency over difficulty: a Fibonacci substitution, a three-site cluster, and a pair of growth runs differing only in their alphabet.

None of this is new physics, and the lattice half of the formalism is due to others. What is attempted is a placement: saying which component of a grammar each part of the phenomenon occupies, and checking that the placement survives contact with small examples.

A word on the standing of each. The lattice-scale material is established and is cited in place of being re-derived; the contribution there is presentational, fixing an example so that the comparison in §4 can be made. The claims about the morphological scale and the asymmetry are arguments about the formalism. The coupling is supported by computations reported here at small cluster sizes and by published simulations at large ones, and §6 separates these and records an implementation error made in the course of the work, together with the reason the validation checks that were passed failed to detect it.

§1 Physical Preliminaries

This section fixes the physical content the rest of the paper assumes. Everything in it is standard, and it is included so that the grammatical claims can be stated without implicit appeal to unstated facts about crystallisation.

1.1 Nucleation and the critical size

A liquid held below its melting temperature may persist without freezing. The crystal is the state of lower free energy, and the delay has a definite cause: forming a crystalline region creates an interface, and the interface costs energy. For a spherical region of radius $r$ the free-energy change is a difference of two terms with different scaling,
$$\Delta G(r) ;=; \underbrace{4\pi r^{2}\gamma}{\text{interface}} ;-; \underbrace{\tfrac{4}{3}\pi r^{3},\Delta g{v}}{\text{bulk}}, \tag{1}$$
with $\gamma>0$ the interfacial tension and $\Delta g{v}>0$ the free energy released per unit volume of liquid converted. The quadratic term dominates at small $r$ and the cubic term at large $r$, so $\Delta G$ rises, turns, and falls. Its maximum sits at the critical radius
$$r_{c} ;=; \frac{2\gamma}{\Delta g_{v}}, \qquad \Delta G^{*} ;=; \Delta G(r_{c}) ;=; \frac{16\pi\gamma^{3}}{3,\Delta g_{v}^{2}} . \tag{2}$$

Remark 1.1 (Threshold character and instantaneous determination). Two consequences of (2) are used below. First, $r_{c}$ is a threshold: a region smaller than $r_{c}$ shrinks and one larger grows, so the transition is governed by whether a fluctuation is large enough and never by how long a small one persists. Second, $r_{c}$ and $\Delta G^{*}$ are computed from a competition of energies at a single instant, with no reference to how the region was assembled. Classical nucleation theory is in this sense a theory of states, and §3 concerns exactly what it leaves undetermined.

1.2 Growth, and the two rate-limiting regimes

Once a region exceeds $r_{c}$ it grows by accretion, and the character of the growth depends on which step is slow. Let $\tau_{D}$ be the time for a molecule to reach the growth front and $\tau_{A}$ the time to attach once it has arrived.

When $\tau_{A}\gg\tau_{D}$ transport is fast, the concentration around the crystal is nearly uniform, and a molecule that fails to attach at one site samples many others. Growth is then governed by interfacial energetics and the result is compact. This is the attachment-limited regime.

When $\tau_{D}\gg\tau_{A}$ transport is slow, a molecule attaches essentially where it first arrives, and the concentration field develops steep gradients. This is the diffusion-limited regime, and it is the one this paper treats.

1.3 Harmonic measure and screening

In the diffusion-limited regime the concentration $u$ of growth units outside the cluster satisfies the quasi-static problem
$$\nabla^{2}u = 0 \ \text{ outside } C, \qquad u \ \text{fixed on } \partial C \text{ and at infinity}, \tag{3}$$
and the local growth velocity is proportional to $\partial u/\partial n$.

Definition 1.2 (Harmonic measure). The harmonic measure of a boundary site is the probability that a random walker released far from the cluster first contacts the cluster at that site. It is the discrete counterpart of the normal gradient of the solution to (3), and it is the growth probability of that site.

Remark 1.3 (Screening). For a harmonic function the normal gradient is largest where the boundary is most convex. A protrusion therefore intercepts a disproportionate share of the arriving flux, while a site in a concavity is reached only by walkers that survive many chances to attach on the way in and is reached exponentially rarely. Growth advances at the tips and stalls in the interior. This effect is called screening, and it is the mechanism behind every morphological claim in this paper.

Screening is a positive feedback on shape, and it is opposed by interfacial tension, which through the Gibbs–Thomson effect penalises exactly the sharpest features. The competition selects a finite feature size, excluding both the smooth sphere and the arbitrarily fine dendrite.

Remark 1.4 (Irreversibility). Accretion in this regime is treated as permanent: an attached unit does not detach. The system therefore does not sample configurations and does not relax toward a minimum, so the final object solves no variational problem over final states. Its form is a record of the order in which material arrived. This is the physical fact that §3 renders grammatically.

Remark 1.5 (Fractal dimension). A cluster produced this way is statistically self-similar over a range of scales, and its mass within radius $R$ obeys $M(R)\sim R^{D}$ for an exponent $D$ strictly between $1$ and $2$ in the plane. Equivalently, covering the cluster with boxes of side $\epsilon$ requires $N(\epsilon)\sim\epsilon^{-D}$ of them. Since the mean density inside radius $R$ scales as $R^{D-2}$, such a cluster becomes arbitrarily sparse as it grows. The accepted value for planar diffusion-limited aggregation in the continuum is $D\approx1.71$.

1.4 Lattices and anisotropy

A crystalline material has a discrete symmetry, and a growth model on a lattice inherits a preferred set of directions from it. Two devices from the simulation literature are used in §5. Restricting growth to a set $\mathcal{D}$ of lattice directions imposes the symmetry on the growth process. Noise reduction requires a site to be selected $H$ times before it is occupied, which averages over the fluctuations of the growth measure and lets a weak directional preference become visible at cluster sizes that are reachable in practice.

§2 The Lattice Scale as a Configuration Language

2.1 Definition of the configuration language

Definition 2.1 (Configuration language). Let $\mathfrak{A}$ be a finite alphabet whose letters name local structural units, and let $\mathfrak{A}^{}$ be the free monoid over $\mathfrak{A}$. A configuration language is a subset $\mathcal{L} \subseteq \mathfrak{A}^{}$ consisting of the admissible configurations of the material, generated by a phrase-structure grammar $\mathcal{G} = (\mathfrak{Q}, \mathfrak{A}, P, S)$ with non-terminals $\mathfrak{Q}$, production set $P$, and start symbol $S$.

Remark 2.2 (Space-group constraints as productions). For a periodic crystal, the constraints defining $\mathcal{L}$ are those of the space group: translational periodicity, point-group symmetry, and occupancy of Wyckoff positions. These are restrictions on which strings are admissible, and they are order-independent. The grammar accepts or rejects a configuration; it says nothing about the sequence by which the configuration came to be assembled.

Example 2.3 (Fibonacci chain). The one-dimensional quasilattice is generated by the D0L system with alphabet ${\mathbb{L}, \mathbb{S}}$, axiom $\mathbb{L}$, and substitution
$$\phi(\mathbb{L}) = \mathbb{L}\mathbb{S}, \qquad \phi(\mathbb{S}) = \mathbb{L},$$
whose language ${\phi^{n}(\mathbb{L})}_{n \in \mathbb{N}}$ is context-free, computed by the grammar with productions $S \to A$, $A \to AB$, $B \to A$, $A \to \mathbb{L}$, $B \to \mathbb{S}$. Inflation of the quasilattice is the endomorphism $\phi$.

2.2 The Fibonacci substitution as a minimal instance

Example 2.4 (The Fibonacci grammar in full). Take $\mathcal{G}_{\rm micro}$ with terminals $\mathfrak{A}={\mathrm{L},\mathrm{S}}$, one non-terminal, axiom $\mathrm{L}$, and the two productions
$$\mathrm{L};\to;\mathrm{L},\mathrm{S}, \qquad \mathrm{S};\to;\mathrm{L}.$$
Iterating from the axiom gives $\mathrm{L}$, $\mathrm{LS}$, $\mathrm{LSL}$, $\mathrm{LSLLS}$, $\mathrm{LSLLSLSL}$, $\mathrm{LSLLSLSLLSLLS}$, and so on. Assign length $\varphi$ to $\mathrm{L}$ and $1$ to $\mathrm{S}$, with $\varphi=(1+\sqrt5)/2$, and read the word as a point set on a line.

Proposition 2.5 (Quasiperiodic order of the generated point set). For the grammar of Example 2.4: the word lengths satisfy $|w_{n+2}|=|w_{n+1}|+|w_n|$; the tile ratio satisfies $#\mathrm{L}/#\mathrm{S}\to\varphi$; and the associated point set has pure-point diffraction with peak positions indexed by $\varphi$.

Verification. Each $\mathrm{L}$ contributes one $\mathrm{L}$ and one $\mathrm{S}$ to the next word and each $\mathrm{S}$ contributes one $\mathrm{L}$, so the letter counts obey the Fibonacci recursion and the ratio converges to the dominant eigenvalue $\varphi$ of $\bigl(\begin{smallmatrix}1&1\1&0\end{smallmatrix}\bigr)$. Direct computation to depth $12$ gives the word lengths
$$1,;2,;3,;5,;8,;13,;21,;34,;55,;89,;144,;233,$$
a ratio $#\mathrm{L}/#\mathrm{S}=233/144=1.618056$ against $\varphi=1.618034$, and a structure factor whose strongest peaks stand in ratio $1.617$, consistent with $\varphi$. Figure 1 displays all three. $\blacksquare$

Figure 1. The micro grammar of Example 2.4. (a) Six steps of the derivation, with $\mathrm{L}$ tiles in blue and $\mathrm{S}$ tiles in red; each row is one application of the two productions to every letter of the row above. (b) The tile ratio converges to $\varphi$: the language fixes this number, and no derivation order enters. (c) The diffraction of the depth-$12$ word is pure point, which is the signature of quasiperiodic order.

Remark 2.6 (Order-independence of the generated quantities). Every quantity in Proposition 2.5 is a property of the set of words the grammar admits. Applying the productions in a different order across the letters of a word changes nothing, since the substitution acts on all letters simultaneously and the result is the same string. This is the sense in which the physics sits in the language.

2.3 Prior results on aperiodic order and their attribution

Remark 2.7 (Attribution). The following are due to others and are used without re-derivation. Fernandes and Marcolli construct a category of context-free languages whose morphisms are rational transductions, exhibit D0L systems as a special case of those morphisms, and give a functor to aperiodic spin chains, extended to multiple context-free grammars and applied to the Korepin integrable model on the icosahedral quasicrystal via a grammar encoding the Ammann planes quasilattice. Three consequences of that work are used below.

First, block-spin renormalisation is a morphism in the grammar category. Coarse graining, which on the physical side relates descriptions at different length scales, has an exact grammatical counterpart. §5 relies on this.

Second, the Chomsky level required is dimension-dependent: one-dimensional classical spin Hamiltonians are modelled by context-free grammars, while two and higher dimensions require context-sensitive grammars.

Third, the level required is also presentation-dependent within a fixed geometry. The inflation rules of the Socolar–Steinhardt and Danzer tilings are context-free, while those of the Ammann rhombohedral tiling are context-sensitive, for the same three-dimensional icosahedral structure.

Remark 2.8 (Status of the lattice-scale material). Remark 2.7 settles the status of the lattice scale. A grammatical description of periodic and aperiodic order exists, is rigorous, and is categorical. This paper claims nothing new there, and §2 is present to fix notation and to make the comparison in §4 possible.

§3 The Morphological Scale as a Derivation

3.1 Definition of the aggregation grammar

Definition 3.1 (Aggregation grammar). Let $\Lambda$ be a lattice with permitted direction set $\mathcal{D} \subseteq \Lambda$, and let a cluster be a finite connected $C \subseteq \Lambda$. An aggregation grammar has a single production schema
$$C ;\longrightarrow; C \cup {v}, \qquad v \notin C, \quad v = u + d \text{ for some } u \in C,; d \in \mathcal{D},$$
applicable at any site $v$ adjacent to $C$ along a permitted direction. A derivation is a sequence $C_{0} \subset C_{1} \subset \cdots \subset C_{N}$ of single-site extensions from a seed $C_{0}$.

Remark 3.2 (Triviality of the production set). Definition 3.1 has essentially one production. The morphological grammar’s content therefore cannot lie in its production set, which is the reverse of the lattice case, where the production set carries the space-group constraints and the derivation is immaterial. What distinguishes one cluster from another is entirely the sequence in which the single production was applied, and at which sites.

Definition 3.3 (Growth measure). Let $u$ be the harmonic function exterior to $C$ with $u$ fixed on $\partial C$ and at infinity. The growth measure $\mu_{C}$ assigns to each admissible site $v$ the probability that a random walker from infinity first contacts $C$ at $v$. A derivation step selects $v$ with probability $\mu_{C}(v)$.

Proposition 3.4 (Nonlocality of the selection rule). $\mu_{C}$ depends on the entire cluster $C$, not on a bounded neighbourhood of $v$. Consequently the aggregation grammar is not context-free in the derivation: the probability of applying the production at a site is a functional of the whole string derived so far.

Argument. $\mu_{C}$ is the harmonic measure of $\partial C$, determined by the solution of an exterior Dirichlet problem whose boundary condition is the whole of $\partial C$. Altering $C$ at any distance changes $u$ everywhere, hence changes $\mu_{C}(v)$ for every $v$. A context-free production is applicable on the basis of a single non-terminal independent of its surroundings, so no context-free schema reproduces this dependence. $\blacksquare$

Remark 3.5 (Agreement with the dimension-dependence of the lattice scale). Proposition 3.4 agrees in direction with the second consequence recorded in Remark 2.7, that two and higher dimensions require context-sensitive grammars. Here the source of the context sensitivity is identified concretely: it is the nonlocality of the harmonic measure. The agreement is qualitative, and the present paper does not establish that the aggregation grammar sits at any particular level of the hierarchy, only that it lies above context-free.

3.2 Irreversibility and the failure of configuration weights

Proposition 3.6 (Order-dependence of the growth measure). Let $v$ and $w$ be admissible sites for a cluster $C$ with $v \ne w$. In general $\mu_{C \cup {v}}(w) \ne \mu_{C}(w)$, so the two orders of application of the production yield different probabilities for the resulting configuration. The derivation order therefore carries physical content, beyond any presentational role.

Proof. Adding $v$ alters the boundary of the exterior Dirichlet problem and hence alters the harmonic measure at every remaining site, including $w$; the inequality is strict whenever $v$ screens or exposes $w$. Since the same final configuration $C \cup {v,w}$ is reached by either order with unequal probability, the probability of a configuration depends on more than the configuration alone. $\blacksquare$

Corollary 3.7 (Absence of a configuration-level generating function). There is no assignment of weights to configurations alone that reproduces the statistics of the aggregation grammar. Any generating description must be indexed by derivations.

Remark 3.8 (Difference from the equilibrium partition function). Corollary 3.7 marks the separation from the lattice scale. There a configuration carries a Boltzmann weight determined by the configuration itself, and the partition function sums over configurations with no reference to how they were reached. Here the corresponding sum requires derivations as its index set. This is the formal content of the observation that a diffusion-limited cluster records its own history.

3.3 Screening on a three-site cluster

Example 3.9 (A three-site cluster with two admissible sites). Let $\Lambda=\mathbb{Z}^{2}$ with $\mathcal{D}$ the four nearest-neighbour directions, and let $C={(0,0),(0,1),(0,2)}$, a vertical bar. Two admissible sites are the tip $v=(0,3)$ and the flank $w=(1,1)$. The grammar offers the same production at both, and the derivation is fixed by which one fires.

Proposition 3.10 (Measured harmonic measure of tip and flank). For Example 3.9, estimating the harmonic measure by random walkers from a distant circle gives
$$\mu_{C}(v) = 0.154, \qquad \mu_{C}(w) = 0.099, \qquad \mu_{C\cup{v}}(w) = 0.084 .$$
Firing the production at the tip therefore lowers the probability of the flank by $15.5%$, and the tip is favoured over the flank by a factor $1.55$ before either fires.

Remark 3.11 (Order-dependence exhibited numerically). The three numbers of Proposition 3.10 exhibit the order dependence directly: $\mu_{C\cup{v}}(w) \ne \mu_{C}(w)$, so the two orders of firing reach the configuration $C\cup{v,w}$ with different probabilities, and no weight on configurations reproduces this. The same asymmetry, compounded over many firings, produces the branched clusters of Figure 2: a site that fires early suppresses its neighbours permanently, and the voids so created persist to the end of the derivation.

Figure 2. One derivation under $\mathcal{G}_{\rm macro}$, whose production set contains the single schema $C \to C \cup {v}$. Colour encodes the derivation index, so the figure is a picture of the derivation, going beyond the final configuration. The branched form is present early and is preserved: by Proposition 3.10 an early firing suppresses its neighbours, so the interior voids visible at $N=200$ are still voids at the end.

§4 Asymmetry Between Language and Derivation

Claim 4.1 (Placement of the physical content). The two scales of crystal formation place the physical content in different components of the grammar. At the lattice scale the content is in the language: which configurations are admissible, order-independent and energetically selected. At the morphological scale the content is in the derivation: the order of application, path-dependent and without variational characterisation. A single formalism covers both scales only when it distinguishes these components, and the observation that each scale admits a grammar leaves the difference between them unstated.

	Lattice scale	Morphological scale
Grammatical object	the language $\mathcal{L}$	the derivation $C_{0} \subset \cdots \subset C_{N}$
Production set	carries the constraints	nearly trivial (Def. 3.1)
Order of application	immaterial	constitutive (Prop. 3.6)
Selection	energetic, variational	harmonic measure, nonlocal
Reached by	many histories	one history
Weight assignable to	configurations	derivations only (Cor. 3.7)
Shared across specimens	yes, exactly	no, in any branch

Remark 4.2 (Reproducibility of symmetry, branch, and dimension). Claim 4.1 predicts a split in what is reproducible. Lattice symmetry is identical in every specimen because it is a property of the language. Branch structure differs in every specimen because it is a property of one derivation. The fractal dimension sits between: it is a statistical property of the ensemble of derivations, and is therefore reproducible across specimens while remaining a property of the growth process and not of any energy function.

§5 Coupling Through the Applicability Condition

§2 and §3 treat the two scales separately. They are not independent, and the mechanism of their coupling has a grammatical form.

Definition 5.1 (Anisotropic restriction). In Definition 3.1, let $\mathcal{D}$ be the nearest-neighbour direction set of the lattice $\Lambda$. The single production is then applicable only at sites offset from the cluster along $\mathcal{D}$. Lattice symmetry thereby enters the morphological grammar as a restriction on where the production may fire.

Remark 5.2 (Correspondence with the simulation procedure). Definition 5.1 coincides with how anisotropy is imposed in the simulation literature. Goold, Somfai and Ball introduce anisotropy by restricting growth to a set of preferred directions, adding prospective growth sites offset from each grown site along each lattice direction. The grammatical description and the numerical procedure are the same operation.

Remark 5.3 (Noise reduction as an averaging device). A restriction on permitted directions competes with the fluctuations of the growth measure, and at small cluster sizes the fluctuations dominate. The standard device for exposing the restriction is noise reduction: a site is occupied only after it has been selected $H$ times, which averages over the diffusion field. Meakin showed that noise reduction on lattices with $n$-fold symmetry, $n \le 6$, produces clusters with $n$ distinct arms resembling much larger clusters grown without it. Ball and Somfai record that small clusters appear robust to the bias of the square lattice while large or noise-reduced clusters are driven to a four-fingered dendrite.

Remark 5.4 (Renormalisation in both formalisms). The coupling has a counterpart in each formalism. On the physical side, Goold, Somfai and Ball characterise anisotropic growth by angular harmonic functions $A_{4}, A_{6}$, find stable fixed points for simple cubic, body-centred cubic and face-centred cubic restrictions, and read the flow in the $(A_{4}, A_{6})$ plane as a renormalisation flow in length scale. On the grammatical side, block-spin renormalisation is a morphism in the category of Remark 2.7. Coarse graining is thus represented in both descriptions, which is what makes a single multi-scale treatment coherent beyond the merely notational.

5.1 Two alphabets under a common production

Example 5.5 (The coupled grammar for two alphabets). Let $\mathcal{G}{\rm int}$ be the pair $(\mathcal{G}{\rm micro}, \mathcal{G}{\rm macro})$ coupled by the single stipulation that the direction set $\mathcal{D}$ of Definition 3.1 is the nearest-neighbour set of the lattice whose symmetry $\mathcal{G}{\rm micro}$ encodes. Two instances are compared:
$$\mathcal{D}{4} = {\pm e{1}, \pm e_{2}}, \qquad \mathcal{D}{6} = {\pm e{1}, \pm e_{2}, \pm(e_{1}-e_{2})},$$
the square and triangular cases. Everything else in the two runs is identical: the same production, the same growth measure, the same seed.

Remark 5.6 (The sticky-site growth rule). The production is implemented in the sticky-site form: when a site is grown, prospective sites are created offset from it along each direction of $\mathcal{D}$, and a prospective site is occupied once it has accumulated $H$ walker contacts. The parameter $H$ is noise reduction in the sense of Remark 5.3. Depositing at the walker’s own position over-weights sites with several occupied neighbours and is known to generate spurious diagonal anisotropy; the sticky-site form avoids this.

Proposition 5.7 (Dependence of arm count on the alphabet). For Example 5.5 at $N=1500$, the anisotropy functions take the values

$H=1$ $H=20$

$\mathcal{D}{4}:\ A{4}$ $+0.07$ $+0.77$

$\mathcal{D}{6}:\ A{6}$ $+0.18$ $+0.40$

In both cases $A_{n}>0$ and increases with noise reduction, so the arms align with the directions of $\mathcal{D}$, and the number of arms equals $|\mathcal{D}|/2$.

	$H=1$	$H=20$
$\mathcal{D}{4}:\ A{4}$	$+0.07$	$+0.77$
$\mathcal{D}{6}:\ A{6}$	$+0.18$	$+0.40$

Remark 5.8 (Interpretation of the measured values). Proposition 5.7 is the coupling in its most direct form. A single discrete datum of the micro grammar, the set $\mathcal{D}$, determines a macroscopic shape, and it does so without altering the production, the growth measure, or any parameter of the derivation. The measured $A_{n}>0$ agrees in sign and in arm count with the published results of Remark 5.3. Figure 3 shows the four clusters.

Figure 3. The integrated grammar of Example 5.5. Top row $\mathcal{D}{4}$, bottom row $\mathcal{D}{6}$; left column $H=1$, middle column $H=20$, right column the corresponding anisotropy function. At $H=1$ the fluctuations of the growth measure dominate and the restriction is barely visible. At $H=20$ the averaging exposes it, and the cluster carries $|\mathcal{D}|/2$ arms aligned with $\mathcal{D}$. The micro alphabet is the only difference between the rows.

5.2 Displacement of the fractal dimension

The strongest available evidence for the coupling is that the restriction moves a macroscopic exponent.

Remark 5.9 (The rigorous upper bound on arm growth). For two-dimensional DLA, Kesten proved that the arms grow at most as $n^{2/3}$ in the cluster mass $n$, which corresponds to $D = 3/2$. This is essentially the only rigorous theorem about the standard model. No non-trivial lower bound is known, and it remains formally open to exclude convergence to a ball, though simulation excludes it.

Remark 5.10 (Published measurement of the dimension drift). Loh, in bias-free simulations of $10^{8}$ particles on lattices of $65536^{2}$ sites with systematic errors reduced below $10^{-12}$, verifies that lattice DLA grows into anisotropic shapes dictated by the anisotropy of the aggregation process, and that the fractal dimension evolves from the continuum value $D \approx 1.71$ for small disc-shaped clusters toward $D = 3/2$ for highly anisotropic clusters with long protruding arms.

This is the quantitative form of the coupling. A restriction on which productions may fire, imposed at the lattice scale, shifts a statistical observable of the derivation ensemble at the morphological scale, and shifts it toward the value fixed by Remark 5.9. The endpoint of the drift is a proved bound, with no fitted constant involved.

Remark 5.11 (Nonuniversality of the on-lattice exponent). Menshutin and collaborators record that on-lattice DLA is lattice dependent and therefore nonuniversal, that its scaling is not governed by a single exponent, and that measured exponents drift with particle number in a way suggesting a transient regime. The coupling claim must accordingly be stated without treating $D$ as a universal constant for the on-lattice case. What is claimed is that the restriction moves the exponent in a definite direction toward a proved bound, and not that a single number characterises either endpoint.

Corollary 5.12 (The coupling in summary form). Under Definition 5.1 the lattice grammar restricts the morphological grammar at the level of production applicability. By Remarks 5.3 and 5.10 that restriction is expressed macroscopically in the arm count of the cluster and in the value of $D$. The two scales are therefore coupled through the applicability condition of a single production, which is the grammatical locus of the coupling.

§6 Standing of the Evidence and Open Questions

Remark 6.1 (Standing of each class of claim). The claims rest on sources of differing weight, and the division should be explicit.

Claim 4.1 and Propositions 3.4–3.6 are arguments about the formalism and require no simulation.

The lattice-scale material of §2 is due to others and is cited; Example 2.4 and Proposition 2.5 are standard facts about the Fibonacci substitution, recomputed here to fix the example.

Propositions 3.10 and 5.7 are computed for this paper. They are small: the first is a harmonic-measure estimate on a three-site cluster, the second a pair of runs at $N=1500$. They establish the sign and the arm count of the coupling and settle no exponent.

The displacement of the fractal dimension in §5.2 rests entirely on the published record, specifically Meakin on noise reduction and arm counts, Ball and Somfai on the onset of lattice-driven dendrites, Goold, Somfai and Ball on anisotropy fixed points, Menshutin and collaborators on nonuniversality, and Loh on the drift of $D$ toward the Kesten bound. Nothing computed here reaches the cluster sizes those results require.

Remark 6.2 (An implementation error and its correction). A first attempt at the numerical check of §5.1 failed, and the episode is reported because it bears on how such checks should be read.

The initial implementation deposited at the walker’s own site whenever any permitted step contacted the cluster. The anisotropy measure was validated against an exact case, returning $A_{n}=1$ for a perfect $n$-armed star, and the contact geometry was instrumented, confirming that every deposition was orthogonal. Nonetheless $A_{4}$ ran from $-0.120$ at $N=10^{3}$ to $-0.144$ at $N=1.2\times10^{4}$ under strong noise reduction, converging to a state whose four arms lay between the lattice axes, with none along them. The sign is opposite to the published behaviour recalled in Remark 5.3.

The cause lies in the growth rule, with cluster size playing no part. Diagonal anisotropy under high noise reduction is a documented artefact of neighbour-dependent occupation rules: Alves and Ferreira showed that such rules yield diagonal patterns in the large-scale and high-noise-reduction limits, and that the pathology is controlled by growth probabilities $P_{k}=(k/n)^{\nu}$, axial below a critical $\nu$ and diagonal above it. Depositing at the walker position over-weights sites with several occupied neighbours and reproduces it. Replacing that rule by the sticky-site form of Remark 5.6 reverses the sign, giving the values of Proposition 5.7.

Two points follow. The validation checks that were passed, an exact-case calibration of the measure and an audit of the contact geometry, were together insufficient to detect the error, since both concern the instrument and the attachment step while the fault lay in the choice of growth site. And a simulation agreeing with the literature in sign and arm count, as the corrected one does, is evidence about the implementation before it is evidence about the claim.

Remark 6.3 (Open questions). Three items are named. The level of the aggregation grammar in the Chomsky hierarchy is not determined here; Proposition 3.4 places it above context-free and nothing further. The relation between the renormalisation flow of Remark 5.4 and the grammatical morphism of Remark 2.7 is stated as a correspondence of roles and not proved to be a correspondence of structures. And the quantitative form of the coupling is at present a single exponent drift, so a sharper statement would require the dependence of $D$ on the restriction to be characterised, with its sign alone being insufficient.

中文

一种关于晶体形成的多尺度生成语法

点阵、形貌，以及两者之间的耦合

黄万宏 · 独立研究者 · huangwanhong@serendip.ngo

摘要

关于物理形成的形式模型通常是关于状态的模型：一个构型被赋予一个能量，而理论在诸构型之间选择。不可逆且历史依赖的诸过程抗拒这一处理，因为它们的结果由它们如何被产生所区分、而不由关于那个产物本身的任何东西所区分。晶体形成同时含有两种。它的点阵由能量选择、并在每一个样本中都相同；它的分枝形貌由累积的次序选择、并独一无二于一段历史。本文以单一的形式体系，即生成语法，为两者建模，并把它们之间的差别定位在那个形式体系之内。

提供三点观察。第一，一个动力学过程可被写作一个生成语法，其中推导承载物理，这使程序性内容成为一个形式对象，并使相关联的诸定理进入可及范围。第二，晶体形成的两个尺度看来占据一个语法的不同组分：点阵尺度是一个关于语言的陈述，即次序无关且由能量选择，而形貌尺度是一个关于推导的陈述，即路径依赖且不容许任何变分刻画；那么仅施于诸构型之上的权重看来不足以复现生长的统计。第三，两个尺度在该形式体系的一个单一点上耦合，即一条产生式的适用条件：点阵字母表限制生长可发生于何处。此处所报告的小计算与已发表的模拟文献一致，即这一限制固定团簇的臂数与取向，而那一文献报告它把分形维数从连续介质值移向一个严格的界。

三个模型中的每一个都被明确给出，连同它的产生式规则、一个实做的例子，以及一幅图。点阵尺度的形式体系归功于他人；此处所尝试的是对各部分的一个安置。

关键词： 生成语法；晶体生长；替换规则；扩散限制凝聚；分形维数。

说明。 这是一篇草稿与讨论论文。反对、讨论与更正欢迎寄至上面的地址。

引言

一个晶体有一个重复单元与一个形状。那个重复单元由能量学固定，并在该物质的每一个样本中被共享。那个形状则不然：两片点阵对称相同的雪花在任何一根分枝上都不一致。

统计物理的标准装置处理这两者中的第一个、而对第二个保持沉默，出于一个结构性的缘由。那个装置给一个构型赋予一个权重并对诸构型求和，故它所计算的一切都是那个产物的一个函数。在一个过程可逆且趋于平衡之处这不丢失任何东西，因为系统采样它的诸构型而历史被冲刷掉。在累积不可逆、到达的材料从不脱离之处，系统什么也不采样，而两个作为构型不可区分的对象，在它们形成的概率上可能曾极其不相等。一个只索引诸构型的理论无法陈述这一差别，而成核的经典处理，即以单一时刻能量的一个局部竞争来计算一个临界尺寸与一个激活势垒，对它也没有任何资源。

所要求的是这样一个形式体系，其中程序是一个头等对象，承载它自己的结构与它自己的诸定理，与它所生成的产物并列。

生成语法恰好把必须被区分的诸组分分开。一个语法由一个产生式集、一个语言与一个推导构成。语言是该语法所接纳之诸对象的集合，是该语法的一个性质。一个推导是穿过诸产生式的一条路径，是单一一段历史的一个性质。两个推导可能终止于同一个字符串。

四个特征使这不止于一次记号的改变。诸产生式是一个有限的、可检视的清单，故对该过程的一个约束成为一个可被编辑、被比较、被推理的句法对象。语言与推导之间的区分在该形式体系中已然是承重的，故一个物理主张可被指派给两者之一，而这一指派携带后果。乔姆斯基层级供出一个关于一个过程所要求之资源的量度，这是对那个过程本身的一个有意义的分类。而诸语法的复合是被充分理解的，这正是一个多尺度说明所要求的：一个长度尺度上的描述同另一个长度尺度上的描述之间的关系，在诸语法之间的态射中有一个对应物，块自旋重整化出现在它们之中。

该形式体系已被应用于周期与非周期序，在那里它是严格且范畴式的。那一工作处理平衡结构，而它指派给一个物理系统的诸对象是语言。程序性的一侧未曾受到可比的处理，而它正是本文诸问题所落之侧。

本文提供四样东西，无一是大的。

一个生长过程的语法呈现（§3）。 扩散限制生长被写作一个带单一产生式的语法，其适用性由调和测度支配。在这一呈现中产生式集近乎平凡、而推导承载内容，这以语法的术语重述了那个熟悉的观察，即这样一个团簇记录它自己的历史。两个小命题伴随它：那个选择规则是非局部的，这把该语法置于上下文无关之上、而不固定它的层级；而一个构型的概率依赖于多于该构型的东西，故仅施于诸构型之上的权重看来不足以复现那个统计。
两个尺度之间的一个不对称（§4）。 那个提示是，晶体形成的两个尺度把它们的物理内容置于该语法的不同组分之中，即点阵尺度处的语言与形貌尺度处的推导次序。这一观察与那个熟悉的、关于什么跨样本可复现的分裂相一致，而它为那一分裂提供一套语汇；并未确立那套语汇买来超出描述的任何东西。
定位在一条产生式处的一个耦合（§5）。 点阵字母表可被读作对生长产生式可在何处激发的一个限制。此处所报告的小计算与那一限制固定团簇的臂数与取向相一致，与已发表的模拟文献一致；那个进一步的主张，即那一限制把分形维数移向凯斯滕所证的界，取自那一文献，且超出此处所计算之任何东西的可及范围。
实做的例子。 三个语法中的每一个都被给出，连同它的诸产生式、一个被完整算出的小例子，以及一幅图。这些例子因透明胜于难度而被选取：一个斐波那契替换、一个三位点团簇，以及一对仅在其字母表上有别的生长运行。

这些没有一样是新物理，而该形式体系的点阵一半归功于他人。所尝试的是一个安置：说出该现象的每一部分占据一个语法的哪一个组分，并检查那一安置经受住与小例子的接触。

关于每一样东西之地位的一个词。点阵尺度的材料是既立的，被引用以代重新推导；那里的贡献是呈现性的，固定一个例子以使§4中的比较可被作出。关于形貌尺度与那个不对称的诸主张是关于该形式体系的论证。那个耦合由此处所报告的小团簇尺寸计算、以及大团簇尺寸的已发表模拟所支持，而§6把这些分开，并记录在工作过程中所犯的一个实现错误，连同那些被通过的验证检查未能侦测到它的缘由。

§1 物理预备

本节固定本文其余部分所假定的物理内容。它其中的一切都是标准的，纳入它是为使那些语法主张可被陈述、而不隐含地诉诸关于结晶的未加陈述之事实。

1.1 成核与临界尺寸

一个被保持在其熔化温度之下的液体可能持续而不冻结。晶体是较低自由能的状态，而这一延迟有一个确定的原因：形成一个晶态区域创造一个界面，而那个界面耗费能量。对一个半径为 $r$ 的球形区域，自由能变化是两个带不同标度之项的差，
$$\Delta G(r) ;=; \underbrace{4\pi r^{2}\gamma}{\text{界面}} ;-; \underbrace{\tfrac{4}{3}\pi r^{3},\Delta g{v}}{\text{体}}, \tag{1}$$
其中 $\gamma>0$ 是界面张力，$\Delta g{v}>0$ 是每单位体积液体被转化所释放的自由能。二次项在小 $r$ 处支配、三次项在大 $r$ 处支配，故 $\Delta G$ 上升、转折、下降。它的极大坐落于临界半径
$$r_{c} ;=; \frac{2\gamma}{\Delta g_{v}}, \qquad \Delta G^{*} ;=; \Delta G(r_{c}) ;=; \frac{16\pi\gamma^{3}}{3,\Delta g_{v}^{2}} . \tag{2}$$

注 1.1（阈值特征与瞬时决定）。 (2) 的两个后果在下面被使用。第一，$r_{c}$ 是一个阈值：一个小于 $r_{c}$ 的区域收缩、一个更大的区域生长，故这一过渡由一个涨落是否足够大所支配、而从不由一个小涨落持续多久所支配。第二，$r_{c}$ 与 $\Delta G^{*}$ 由单一时刻能量的一个竞争所计算，不涉及那个区域如何被组装。经典成核理论在这一意义上是一个关于状态的理论，而§3所关乎的恰恰是它所留下未决之物。

1.2 生长，以及两个速率限制区制

一个区域一旦超过 $r_{c}$ 便通过累积生长，而生长的特征取决于哪一步是慢的。设 $\tau_{D}$ 为一个分子抵达生长前沿的时间，$\tau_{A}$ 为它一旦到达便附着的时间。

当 $\tau_{A}\gg\tau_{D}$ 时输运是快的，晶体周围的浓度近乎均匀，而一个未能在一个位点附着的分子采样许多其他位点。生长于是由界面能量学支配，而结果是紧致的。这是附着限制区制。

当 $\tau_{D}\gg\tau_{A}$ 时输运是慢的，一个分子基本上在它首先到达之处附着，而浓度场发展出陡峭的梯度。这是扩散限制区制，而它是本文所处理的那个。

1.3 调和测度与屏蔽

在扩散限制区制中，团簇之外生长单元的浓度 $u$ 满足准静态问题
$$\nabla^{2}u = 0 \ \text{（在 } C \text{ 之外）}, \qquad u \ \text{在 } \partial C \text{ 上与无穷远处被固定}, \tag{3}$$
而局部生长速度正比于 $\partial u/\partial n$。

定义 1.2（调和测度）。 一个边界位点的调和测度是一个从远离团簇处被释放的随机游走者首次在那个位点接触团簇的概率。它是 (3) 之解的法向梯度的离散对应物，而它是那个位点的生长概率。

注 1.3（屏蔽）。 对一个调和函数，法向梯度在边界最凸之处最大。因而一个凸起截获到达通量中不成比例的一份，而一个处在凹处的位点只被那些在进来的路上挺过许多次附着机会的游走者抵达，并被指数地罕见地抵达。生长在尖端处推进、在内部处停滞。这一效应被称为屏蔽，而它是本文中每一个形貌主张背后的机制。

屏蔽是一个对形状的正反馈，而它被界面张力对抗，界面张力通过吉布斯–汤姆孙效应恰恰惩罚最尖锐的特征。这一竞争选择一个有限的特征尺寸，排除光滑的球与任意精细的枝晶两者。

注 1.4（不可逆性）。 这一区制中的累积被当作永久的：一个附着的单元不脱离。因而系统不采样诸构型、不向一个极小弛豫，故最终对象不解任何关于最终状态的变分问题。它的形式是材料到达之次序的一个记录。这是§3以语法呈现的那个物理事实。

注 1.5（分形维数）。 以这种方式产生的一个团簇在一个尺度范围上是统计自相似的，而它在半径 $R$ 之内的质量对一个在平面中严格介于 $1$ 与 $2$ 之间的指数 $D$ 服从 $M(R)\sim R^{D}$。等价地，以边长为 $\epsilon$ 的盒子覆盖该团簇需要 $N(\epsilon)\sim\epsilon^{-D}$ 个。既然半径 $R$ 之内的平均密度标度为 $R^{D-2}$，这样一个团簇随生长变得任意稀疏。平面扩散限制凝聚在连续介质中的公认值是 $D\approx1.71$。

1.4 点阵与各向异性

一个晶态材料有一个离散的对称，而一个点阵上的生长模型从它继承一组优先的方向。来自模拟文献的两个手法在§5中被使用。把生长限制到一组点阵方向 $\mathcal{D}$，把那个对称强加于生长过程之上。降噪要求一个位点在它被占据之前被选择 $H$ 次，这对生长测度的诸涨落取平均，并让一个弱的方向偏好在实践中可达的团簇尺寸上变得可见。

§2 点阵尺度作为一个构型语言

2.1 构型语言的定义

定义 2.1（构型语言）。 设 $\mathfrak{A}$ 为一个有限字母表，其字母命名局部结构单元，并设 $\mathfrak{A}^{}$ 为 $\mathfrak{A}$ 上的自由幺半群。一个构型语言是一个子集 $\mathcal{L} \subseteq \mathfrak{A}^{}$，由该材料的诸可容许构型构成，由一个短语结构语法 $\mathcal{G} = (\mathfrak{Q}, \mathfrak{A}, P, S)$ 生成，其中有非终结符 $\mathfrak{Q}$、产生式集 $P$，与起始符 $S$。

注 2.2（空间群约束作为诸产生式）。 对一个周期晶体，定义 $\mathcal{L}$ 的诸约束是空间群的诸约束：平移周期性、点群对称，以及维科夫位置的占据。这些是关于哪些字符串可容许的限制，而它们是次序无关的。该语法接受或拒绝一个构型；它对那个构型得以被组装所经由的序列什么也没说。

例 2.3（斐波那契链）。 一维准晶格由带字母表 ${\mathbb{L}, \mathbb{S}}$、公理 $\mathbb{L}$，与替换
$$\phi(\mathbb{L}) = \mathbb{L}\mathbb{S}, \qquad \phi(\mathbb{S}) = \mathbb{L}$$
的 D0L 系统生成，其语言 ${\phi^{n}(\mathbb{L})}_{n \in \mathbb{N}}$ 是上下文无关的，由带产生式 $S \to A$、$A \to AB$、$B \to A$、$A \to \mathbb{L}$、$B \to \mathbb{S}$ 的语法计算。准晶格的膨胀是自同态 $\phi$。

2.2 斐波那契替换作为一个极小实例

例 2.4（完整的斐波那契语法）。 取 $\mathcal{G}_{\rm micro}$，带终结符 $\mathfrak{A}={\mathrm{L},\mathrm{S}}$、一个非终结符、公理 $\mathrm{L}$，与两条产生式
$$\mathrm{L};\to;\mathrm{L},\mathrm{S}, \qquad \mathrm{S};\to;\mathrm{L}.$$
从公理迭代给出 $\mathrm{L}$、$\mathrm{LS}$、$\mathrm{LSL}$、$\mathrm{LSLLS}$、$\mathrm{LSLLSLSL}$、$\mathrm{LSLLSLSLLSLLS}$，以此类推。给 $\mathrm{L}$ 赋长度 $\varphi$、给 $\mathrm{S}$ 赋长度 $1$，其中 $\varphi=(1+\sqrt5)/2$，并把这个词读作一条线上的一个点集。

命题 2.5（所生成点集的准周期序）。 对例 2.4 的语法：词长满足 $|w_{n+2}|=|w_{n+1}|+|w_n|$；瓷砖比满足 $#\mathrm{L}/#\mathrm{S}\to\varphi$；而相关联的点集有纯点衍射，其峰位由 $\varphi$ 索引。

验证。 每一个 $\mathrm{L}$ 向下一个词贡献一个 $\mathrm{L}$ 与一个 $\mathrm{S}$、每一个 $\mathrm{S}$ 贡献一个 $\mathrm{L}$，故字母计数服从斐波那契递归，而那个比收敛到 $\bigl(\begin{smallmatrix}1&1\1&0\end{smallmatrix}\bigr)$ 的主特征值 $\varphi$。直接计算到深度 $12$ 给出词长
$$1,;2,;3,;5,;8,;13,;21,;34,;55,;89,;144,;233,$$
一个比 $#\mathrm{L}/#\mathrm{S}=233/144=1.618056$ 对照 $\varphi=1.618034$，以及一个结构因子，其最强的峰以比 $1.617$ 相立，与 $\varphi$ 一致。图 1 显示所有三者。$\blacksquare$

图 1。 例 2.4 的微观语法。(a) 该推导的六步，$\mathrm{L}$ 瓷砖为蓝色、$\mathrm{S}$ 瓷砖为红色；每一行是两条产生式对上一行每一个字母的一次应用。(b) 瓷砖比收敛到 $\varphi$：语言固定这个数，而没有推导次序进入。(c) 深度 $12$ 之词的衍射是纯点，这是准周期序的标志。

注 2.6（所生成诸量的次序无关性）。 命题 2.5 中的每一个量都是该语法所接纳之词集的一个性质。在一个词的诸字母上以一个不同的次序应用诸产生式什么也不改变，因为那个替换同时作用于所有字母、而结果是同一个字符串。这正是物理坐落于语言之中的意义。

2.3 关于非周期序的先前结果及其归属

注 2.7（归属）。 以下诸结果归功于他人，并被不加重新推导地使用。费尔南德斯与马尔科利构造一个上下文无关语言的范畴，其态射是有理转换，把 D0L 系统展现为那些态射的一个特例，并给出一个到非周期自旋链的函子，扩展到多重上下文无关语法，并经由一个编码阿曼平面准晶格的语法应用于二十面体准晶上的科列平可积模型。那一工作的三个后果在下面被使用。

第一，块自旋重整化是那个语法范畴中的一个态射。粗粒化，在物理一侧它关联不同长度尺度上的诸描述，有一个精确的语法对应物。§5 依赖于此。

第二，所要求的乔姆斯基层级是维度依赖的：一维经典自旋哈密顿量由上下文无关语法建模，而二维及更高维要求上下文相关语法。

第三，在一个固定的几何之内，所要求的层级也是呈现依赖的。索科拉尔–斯坦哈特与丹策尔铺砌的膨胀规则是上下文无关的，而阿曼菱面体铺砌的那些是上下文相关的，对同一个三维二十面体结构。

注 2.8（点阵尺度材料的地位）。 注 2.7 了结点阵尺度的地位。周期与非周期序的一个语法描述存在、是严格的、且是范畴式的。本文在那里不主张任何新东西，而§2在此是为固定记号、并使§4中的比较成为可能。

§3 形貌尺度作为一个推导

3.1 凝聚语法的定义

定义 3.1（凝聚语法）。 设 $\Lambda$ 为一个带许可方向集 $\mathcal{D} \subseteq \Lambda$ 的点阵，并设一个团簇为一个有限连通的 $C \subseteq \Lambda$。一个凝聚语法有一个单一产生式模式
$$C ;\longrightarrow; C \cup {v}, \qquad v \notin C, \quad v = u + d \text{（对某个 } u \in C,; d \in \mathcal{D}\text{）},$$
在任何一个沿一个许可方向邻接于 $C$ 的位点 $v$ 处可适用。一个推导是一个从一个种子 $C_{0}$ 出发的单位点扩展的序列 $C_{0} \subset C_{1} \subset \cdots \subset C_{N}$。

注 3.2（产生式集的平凡性）。 定义 3.1 本质上有一条产生式。因而形貌语法的内容不可能在于它的产生式集，这是点阵情形的反面，在点阵情形中产生式集承载空间群约束、而推导无关紧要。把一个团簇同另一个区分开来的，完全是那单一产生式被应用的序列、以及在哪些位点被应用。

定义 3.3（生长测度）。 设 $u$ 为 $C$ 外部的调和函数，$u$ 在 $\partial C$ 上与无穷远处被固定。生长测度 $\mu_{C}$ 给每一个可容许位点 $v$ 赋予一个从无穷远来的随机游走者首次在 $v$ 处接触 $C$ 的概率。一个推导步骤以概率 $\mu_{C}(v)$ 选择 $v$。

命题 3.4（选择规则的非局部性）。 $\mu_{C}$ 依赖于整个团簇 $C$、而非依赖于 $v$ 的一个有界邻域。因而凝聚语法在推导中不是上下文无关的：在一个位点应用那个产生式的概率是迄今所推导之整个字符串的一个泛函。

论证。 $\mu_{C}$ 是 $\partial C$ 的调和测度，由一个外部狄利克雷问题的解所决定，该问题的边界条件是整个 $\partial C$。在任何距离处改变 $C$ 都在各处改变 $u$，因而对每一个 $v$ 都改变 $\mu_{C}(v)$。一条上下文无关产生式基于一个独立于其周遭的单一非终结符便可适用，故没有任何上下文无关模式复现这一依赖。$\blacksquare$

注 3.5（与点阵尺度之维度依赖性的一致）。 命题 3.4 在方向上与注 2.7 中所记录的第二个后果一致，即二维及更高维要求上下文相关语法。此处那个上下文相关性的来源被具体地辨认：它是调和测度的非局部性。这一一致是定性的，而当下这篇论文并不确立凝聚语法坐落于那个层级的任何特定层级，只确立它位于上下文无关之上。

3.2 不可逆性与构型权重的失败

命题 3.6（生长测度的次序依赖性）。 设 $v$ 与 $w$ 为一个团簇 $C$ 的可容许位点，$v \ne w$。一般地 $\mu_{C \cup {v}}(w) \ne \mu_{C}(w)$，故那个产生式的两个应用次序对由此产生的构型得出不同的概率。因而推导次序承载物理内容，超出任何呈现性的角色。

证明。 添加 $v$ 改变那个外部狄利克雷问题的边界，因而改变每一个剩余位点处的调和测度，包括 $w$；每当 $v$ 屏蔽或暴露 $w$ 时这一不等式是严格的。既然同一个最终构型 $C \cup {v,w}$ 以任一次序被以不相等的概率抵达，一个构型的概率依赖于多于该构型本身的东西。$\blacksquare$

推论 3.7（一个构型层面之生成函数的缺席）。 不存在一个仅施于诸构型之上、复现凝聚语法之统计的权重赋值。任何生成性的描述都必须由诸推导索引。

注 3.8（与平衡配分函数的差别）。 推论 3.7 标记同点阵尺度的分离。在那里一个构型携带一个由那个构型本身所决定的玻尔兹曼权重，而配分函数对诸构型求和、不涉及它们如何被抵达。在这里对应的求和要求诸推导作为它的索引集。这是那个观察的形式内容，即一个扩散限制团簇记录它自己的历史。

3.3 一个三位点团簇上的屏蔽

例 3.9（一个带两个可容许位点的三位点团簇）。 设 $\Lambda=\mathbb{Z}^{2}$，$\mathcal{D}$ 为四个最近邻方向，并设 $C={(0,0),(0,1),(0,2)}$，一根竖条。两个可容许位点是尖端 $v=(0,3)$ 与侧翼 $w=(1,1)$。该语法在两者处都提供同一条产生式，而推导由哪一个激发所固定。

命题 3.10（尖端与侧翼被测得的调和测度）。 对例 3.9，以从一个远圆来的随机游走者估计调和测度给出
$$\mu_{C}(v) = 0.154, \qquad \mu_{C}(w) = 0.099, \qquad \mu_{C\cup{v}}(w) = 0.084 .$$
因而在尖端激发那个产生式把侧翼的概率降低 $15.5%$，而在任一者激发之前尖端以一个因子 $1.55$ 被偏好于侧翼。

注 3.11（数值地展现的次序依赖性）。 命题 3.10 的三个数直接展现那个次序依赖性：$\mu_{C\cup{v}}(w) \ne \mu_{C}(w)$，故两个激发次序以不同的概率抵达构型 $C\cup{v,w}$，而没有任何施于诸构型之上的权重复现这一点。同一个不对称，在许多次激发上复利，产生图 2 的分枝团簇：一个早激发的位点永久地压制它的邻居，而如此创造的诸空洞持续到那个推导的末尾。

图 2。 $\mathcal{G}_{\rm macro}$ 之下的一个推导，其产生式集含有单一模式 $C \to C \cup {v}$。颜色编码推导索引，故该图是那个推导的一幅图像，超出那个最终构型。分枝形式早早在场并被保留：由命题 3.10 一次早激发压制它的邻居，故在 $N=200$ 时可见的内部空洞在末尾时仍是空洞。

§4 语言与推导之间的不对称

主张 4.1（物理内容的安置）。 晶体形成的两个尺度把物理内容置于该语法的不同组分之中。在点阵尺度，内容在于语言：哪些构型可容许，次序无关且由能量选择。在形貌尺度，内容在于推导：应用的次序，路径依赖且无变分刻画。单一的一个形式体系只在它区分这些组分时才覆盖两个尺度，而每一尺度都容许一个语法这一观察，使它们之间的差别未被陈述。

	点阵尺度	形貌尺度
语法对象	语言 $\mathcal{L}$	推导 $C_{0} \subset \cdots \subset C_{N}$
产生式集	承载诸约束	近乎平凡（定义 3.1）
应用次序	无关紧要	构成性的（命题 3.6）
选择	能量的、变分的	调和测度、非局部
被抵达经由	许多段历史	一段历史
权重可赋予	诸构型	仅诸推导（推论 3.7）
跨样本共享	是，精确地	否，在任何分枝上

注 4.2（对称、分枝与维数的可复现性）。 主张 4.1 预测在什么可复现上的一个分裂。点阵对称在每一个样本中都相同，因为它是语言的一个性质。分枝结构在每一个样本中都不同，因为它是一段推导的一个性质。分形维数坐落于两者之间：它是诸推导之系综的一个统计性质，因而跨样本可复现，同时仍是那个生长过程的一个性质、而非任何能量函数的一个性质。

§5 通过适用条件的耦合

§2 与 §3 分别处理两个尺度。它们不是独立的，而它们耦合的机制有一个语法的形式。

定义 5.1（各向异性限制）。 在定义 3.1 中，设 $\mathcal{D}$ 为点阵 $\Lambda$ 的最近邻方向集。那个单一产生式于是只在从团簇沿 $\mathcal{D}$ 偏移的位点处可适用。点阵对称由此作为一个对产生式可在何处激发的限制进入形貌语法。

注 5.2（与模拟程序的对应）。 定义 5.1 与各向异性在模拟文献中被强加的方式相合。古尔德、索姆法伊与鲍尔通过把生长限制到一组优先方向来引入各向异性，从每一个已生长的位点沿每一个点阵方向添加预期的生长位点。那个语法描述与那个数值程序是同一个操作。

注 5.3（降噪作为一个取平均的手法）。 一个对许可方向的限制与生长测度的诸涨落竞争，而在小团簇尺寸处那些涨落支配。暴露那个限制的标准手法是降噪：一个位点只在它被选择 $H$ 次之后才被占据，这对那个扩散场取平均。米金显示，在带 $n$ 重对称（$n \le 6$）的点阵上降噪产生带 $n$ 条不同臂、酷似不带降噪所生长之更大得多团簇的诸团簇。鲍尔与索姆法伊记录，小团簇看来对方形点阵的偏置稳健，而大的或降噪的团簇被驱向一个四指枝晶。

注 5.4（两个形式体系中的重整化）。 那个耦合在每一个形式体系中都有一个对应物。在物理一侧，古尔德、索姆法伊与鲍尔以角向调和函数 $A_{4}, A_{6}$ 刻画各向异性生长，为简单立方、体心立方与面心立方限制找到稳定不动点，并把 $(A_{4}, A_{6})$ 平面中的流读作一个长度尺度上的重整化流。在语法一侧，块自旋重整化是注 2.7 那个范畴中的一个态射。因而粗粒化在两个描述中都被表示，这正是使单一的一个多尺度处理超出仅仅记号地融贯的东西。

5.1 一条共同产生式之下的两个字母表

例 5.5（两个字母表的耦合语法）。 设 $\mathcal{G}{\rm int}$ 为对 $(\mathcal{G}{\rm micro}, \mathcal{G}{\rm macro})$，由单一的规定所耦合，即定义 3.1 的方向集 $\mathcal{D}$ 是其对称由 $\mathcal{G}{\rm micro}$ 编码之点阵的最近邻集。比较两个实例：
$$\mathcal{D}{4} = {\pm e{1}, \pm e_{2}}, \qquad \mathcal{D}{6} = {\pm e{1}, \pm e_{2}, \pm(e_{1}-e_{2})},$$
即方形与三角情形。两个运行中的其他一切都相同：同一条产生式、同一个生长测度、同一个种子。

注 5.6（粘附位点生长规则）。 那个产生式以粘附位点形式被实现：当一个位点被生长时，预期位点被从它沿 $\mathcal{D}$ 的每一个方向偏移地创造，而一个预期位点一旦累积了 $H$ 次游走者接触便被占据。参数 $H$ 是注 5.3 意义上的降噪。在游走者自己的位置沉积会过度加权带数个已占据邻居的位点，且已知会生成虚假的对角各向异性；粘附位点形式避免这一点。

命题 5.7（臂数对字母表的依赖）。 对例 5.5 在 $N=1500$ 处，各向异性函数取值

$H=1$ $H=20$

$\mathcal{D}{4}:\ A{4}$ $+0.07$ $+0.77$

$\mathcal{D}{6}:\ A{6}$ $+0.18$ $+0.40$

在两种情形中 $A_{n}>0$ 并随降噪增加，故诸臂与 $\mathcal{D}$ 的诸方向对齐，而臂的数目等于 $|\mathcal{D}|/2$。

	$H=1$	$H=20$
$\mathcal{D}{4}:\ A{4}$	$+0.07$	$+0.77$
$\mathcal{D}{6}:\ A{6}$	$+0.18$	$+0.40$

注 5.8（对被测值的诠释）。 命题 5.7 是那个耦合最直接的形式。微观语法的一个单一离散数据，即集合 $\mathcal{D}$，决定一个宏观形状，而它这样做不改变那个产生式、那个生长测度，或那个推导的任何参数。被测的 $A_{n}>0$ 在符号与臂数上与注 5.3 的已发表结果一致。图 3 显示那四个团簇。

图 3。 例 5.5 的整合语法。上行 $\mathcal{D}{4}$，下行 $\mathcal{D}{6}$；左列 $H=1$，中列 $H=20$，右列对应的各向异性函数。在 $H=1$ 处生长测度的诸涨落支配、而那个限制勉强可见。在 $H=20$ 处那个取平均暴露它，而团簇带有 $|\mathcal{D}|/2$ 条与 $\mathcal{D}$ 对齐的臂。微观字母表是诸行之间唯一的差别。

5.2 分形维数的位移

对那个耦合可得的最强证据是那个限制移动一个宏观指数。

注 5.9（臂生长的严格上界）。 对二维扩散限制凝聚，凯斯滕证明诸臂在团簇质量 $n$ 中至多如 $n^{2/3}$ 生长，这对应于 $D = 3/2$。这本质上是关于那个标准模型唯一的严格定理。没有非平凡的下界被知晓，而排除向一个球的收敛在形式上仍是悬而未决的，尽管模拟排除它。

注 5.10（维数漂移的已发表测量）。 罗，在 $65536^{2}$ 位点之点阵上 $10^{8}$ 粒子、系统误差被降到 $10^{-12}$ 之下的无偏置模拟中，验证点阵扩散限制凝聚生长成由凝聚过程的各向异性所支配的各向异性形状，而分形维数从小的盘形团簇的连续介质值 $D \approx 1.71$ 演化到带长而凸出之臂的高度各向异性团簇的 $D = 3/2$。

这是那个耦合的定量形式。一个对哪些产生式可激发的限制，在点阵尺度被强加，移动形貌尺度处推导系综的一个统计可观测量，并把它移向注 5.9 所固定的那个值。那个漂移的终点是一个被证的界，不涉及任何被拟合的常数。

注 5.11（在点阵上之指数的非普适性）。 门舒京与合作者记录，在点阵上的扩散限制凝聚是点阵依赖的、因而是非普适的，它的标度不由一个单一指数支配，而被测的诸指数随粒子数以一种提示一个瞬态区制的方式漂移。因而那个耦合主张必须被陈述、而不把 $D$ 当作在点阵上情形的一个普适常数。所主张的是那个限制把那个指数沿一个确定的方向移向一个被证的界，而非一个单一的数刻画任一终点。

推论 5.12（概要形式的耦合）。 在定义 5.1 之下，点阵语法在产生式适用性的层面上限制形貌语法。由注 5.3 与 5.10，那个限制在团簇的臂数中、以及在 $D$ 的值中被宏观地表达。因而两个尺度通过一条单一产生式的适用条件耦合，这就是那个耦合的语法所在。

§6 证据的地位与开放的问题

注 6.1（每一类主张的地位）。 这些主张依据不同分量的来源，而这一划分应予明确。

主张 4.1 与命题 3.4–3.6 是关于该形式体系的论证，无需任何模拟。

§2 的点阵尺度材料归功于他人并被引用；例 2.4 与命题 2.5 是关于斐波那契替换的标准事实，在此被重新计算以固定那个例子。

命题 3.10 与 5.7 是为本文计算的。它们是小的：第一个是一个三位点团簇上的一次调和测度估计，第二个是一对在 $N=1500$ 处的运行。它们确立那个耦合的符号与臂数，且不了结任何指数。

§5.2 中分形维数的位移完全依据已发表的记录，具体是米金关于降噪与臂数、鲍尔与索姆法伊关于点阵驱动之枝晶的出现、古尔德索姆法伊与鲍尔关于各向异性不动点、门舒京与合作者关于非普适性，以及罗关于 $D$ 向凯斯滕界的漂移。此处所计算的没有任何东西抵达那些结果所要求的团簇尺寸。

注 6.2（一个实现错误及其更正）。 §5.1 数值检查的一次首次尝试失败了，而这一插曲被报告，因为它关乎这样的检查应如何被读。

最初的实现在任何一个许可步骤接触团簇时便在游走者自己的位点沉积。那个各向异性度量对着一个精确情形被验证，对一个完美的 $n$ 臂星返回 $A_{n}=1$，而那个接触几何被工具化，确认每一次沉积都是正交的。尽管如此 $A_{4}$ 在强降噪之下从 $N=10^{3}$ 处的 $-0.120$ 跑到 $N=1.2\times10^{4}$ 处的 $-0.144$，收敛到一个其四条臂躺在诸点阵轴之间、没有一条沿它们的状态。这个符号与注 5.3 中所回想的已发表行为相反。

那个原因在于那个生长规则，团簇尺寸不起任何作用。高降噪之下的对角各向异性是邻居依赖之占据规则的一个有记录的人为产物：阿尔维斯与费雷拉显示，这样的规则在大尺度与高降噪极限中产出对角图案，而那个病态由生长概率 $P_{k}=(k/n)^{\nu}$ 所控制，在一个临界 $\nu$ 之下是轴向的、在其之上是对角的。在游走者位置沉积过度加权带数个已占据邻居的位点，并复现它。以注 5.6 的粘附位点形式替换那个规则翻转那个符号，给出命题 5.7 的诸值。

两点随之而来。那些被通过的验证检查，即那个度量的一次精确情形校准与那个接触几何的一次审计，合起来不足以侦测那个错误，因为两者都关乎那个仪器与那个附着步骤、而那个故障在于生长位点的选择。而一个在符号与臂数上与文献一致的模拟，一如那个被更正的模拟所做的，先是关于那个实现的证据、然后才是关于那个主张的证据。

注 6.3（开放的问题）。 三项被命名。凝聚语法在乔姆斯基层级中的层级在此不被决定；命题 3.4 把它置于上下文无关之上、别无其他。注 5.4 的重整化流与注 2.7 的语法态射之间的关系，被陈述为诸角色的一个对应、而未被证明是诸结构的一个对应。而那个耦合的定量形式当前是一个单一指数漂移，故一个更锐利的陈述会要求 $D$ 对那个限制的依赖被刻画，仅有它的符号是不足的。