An experiment sees a Farey sequence

Every fraction holds a window computed a devil’s staircase: particles on a line with any repulsion that is positive, decreasing and convex have, at every rational density, a ground state holding an interval of the chemical potential to itself. Seven hundred and nine steps came out of that computation, the widest taking nineteen per cent of the axis and the narrowest two parts in a million million. It ended on the question that separates a staircase from a phase diagram: which of those steps would anyone ever see?

A step narrower than the fluctuation of the controlling field is never observed, because the field wanders across it before the system settles. A step whose period is longer than the crystal’s coherent domain cannot be told from an incommensurate state, because the domain never holds one full repeat. Both are thresholds, both can be set against the widths, and together they turn a staircase into a count.

The count turns out to be arithmetic, and exact. Every step with the same denominator has the same width. So a resolution does not pick out an irregular scatter of steps: it shows every density whose denominator is at most some bound and not one density beyond it. What an experiment sees is a Farey sequence.

One denominator, one width #

The widths the earlier computation reported were read off the energies directly: the energy per site of the most uniform arrangement at every density with denominator up to forty-eight, and the width of each step as the jump in the slope between its neighbours. Those widths can be compared fraction by fraction, and within a denominator they agree.

For the long-ranged exponential repulsion, the step at one seventh has width 0.11195 in units of the potential, and so do the steps at two, three, four, five and six sevenths, to the fourth significant figure. For the short-ranged exponential they agree to the tenth. The disagreement that remains is not between numerators; it is the amount by which a secant over a finite list of densities overstates the step it approximates, and it shrinks as the list is refined.

The reason is a closed form, and it is due to Bak and Bruinsma, who derived it for this model in 1982. The width of every step with denominator $Q$ is

$$w(Q) = Q \sum_{n \ge 1} n \left[ V(nQ+1) - 2V(nQ) + V(nQ-1) \right],$$

$Q$ times a weighted sum of the repulsion’s second differences at the multiples of $Q$. The numerator does not appear. Computed from this formula and from the energies directly, the widths agree to $4 \times 10^{-10}$ for the short exponential and to $10^{-4}$ for the long one at every denominator up to ten, which is the precision the direct computation carries at those sizes.

The formula also says what convexity buys. Each bracket is a second difference, and a convex repulsion makes every one of them positive; so every step has positive width, which is the staircase. The same formula applied to a repulsion with a well at a distance of two gives a width of −13.5 at denominator three — a negative interval, which is the formula’s way of saying that the uniform arrangement at one third is not the ground state there. The computation refuses to read a staircase off it.

What a resolution shows #

Say an experiment can resolve a step that takes a share $\delta$ of the chemical-potential axis — one part in a thousand, or in a million. It sees the steps wider than $\delta$. Because the width depends on the denominator alone and falls as the denominator grows, those are exactly the steps with denominator up to the largest $Q$ whose width still exceeds $\delta$. Every such fraction is visible; no other is.

At one part in a hundred the long exponential shows 27 steps: every fraction with denominator up to nine. At one part in a thousand, 127: every fraction up to twenty. At one part in ten thousand, 307: every fraction up to thirty-one. Those three numbers are $\sum_{q=2}^{Q} \varphi(q)$ at $Q = 9$, $20$ and $31$ — the number of fractions in lowest terms between nought and one with denominator at most $Q$, which is a sum of Euler’s totient function, the count of numerators coprime to each denominator. The number of phases an experiment sees is a number-theoretic function of one integer, and on a large scale it grows as $3Q^2/\pi^2$.

The same statement read from the energies rather than from the formula holds too: at one part in a hundred and one part in a thousand, the steps the direct computation reports as wider than the resolution are exactly the fractions up to one denominator, for all three repulsions. The visible set is a Farey sequence on either route.

The strips also show what is not visible. At one part in a hundred the axis between the wide blocks is pale — forty-two per cent of it, for this repulsion, which spreads its axis over many denominators. At one part in a thousand the pale share is eight per cent, and at one part in ten thousand one per cent, in slivers. They are not gaps between phases. They are steps too narrow to draw, and the next section adds them up.

The steps are the whole axis #

The chemical potential over which the density lies strictly between nought and one runs from nought to $2 \sum_{n \ge 1} V(n)$: at the bottom the particles are far apart and adding one costs nothing, and at the top the lattice is full and removing one saves its interaction with every other on both sides. Every step lies in that interval. The question the earlier essay could only answer approximately is whether they fill it.

They do. Summing $\varphi(Q)\, w(Q)$ over every denominator — each denominator contributing one step for each numerator coprime to it — gives $2 \sum V(n)$ to fifteen significant figures for the short exponential and the inverse sixth power, and to the same precision for the long exponential once denominators to a few hundred are included. No interval of the chemical potential is left over. Every irrational density is the ground state at exactly one value of the chemical potential and at no interval around it, which is the precise sense in which the staircase is complete.

That settles what the unresolved part of a measurement is. At any resolution the pale remainder of the axis is made of steps too narrow to see — as many of them as there are fractions with larger denominators — and not of incommensurate phases. An experiment that finds a density drifting smoothly with the field has not found an incommensurate ground state; it has found a staircase whose steps are narrower than its resolution. The drift and the staircase are the same observation at two resolutions, which is the difficulty the satellites that need a second integer meets from the diffraction side: a wavevector that seems to vary continuously with temperature may be locking and unlocking at every rational on the way.

How fast the denominator grows is the interaction’s only say #

Everything above holds for every convex repulsion. What differs between repulsions is how fast $w(Q)$ falls, and that decides how many steps a given resolution reaches.

For an exponential repulsion the widths fall by a fixed factor per denominator — $e^{-0.581}$ each step for $e^{-0.6r}$ and $e^{-0.231}$ for $e^{-0.25r}$, measured between denominators thirty and eighty, which is the repulsion’s own decay rate less a slowly vanishing correction from the prefactor. So the largest visible denominator grows with the logarithm of the resolution: for the short exponential every factor of ten in resolution adds four to it, and the count of visible steps runs 21, 45, 79, 139, 199 as the resolution goes from one part in a hundred to one part in a million. Improving an experiment a millionfold, from one part in a hundred to one part in a hundred million, takes it from 21 phases to 343.

For the inverse sixth power the widths fall as a power of the denominator, $Q^{-7.005}$ — one power steeper than the potential. The largest visible denominator then grows as a power of the resolution, by a factor of $10^{1/7} = 1.39$ for each factor of ten. At every practical resolution it shows the fewest steps of the three — three at one part in a hundred, because the step at one half takes ninety-six per cent of the axis on its own — and yet it gains on the exponentials without limit. At one part in ten thousand million it ties with the short exponential at 529 steps each, and past that it shows more.

So which of two materials shows the richer staircase is not a fixed fact about them. It depends on how finely they are looked at, and the answer reverses at a resolution set by where a logarithm and a power cross.

The same count, read in a diffraction pattern #

A locked state is not usually seen as a density on an axis. It is seen as a set of reflections: the chain at density $P/Q$ repeats every $Q$ sites, so its pattern has reflections at multiples of $1/Q$ of the substrate’s reciprocal spacing, and a step is recognised when those reflections stop moving as the field is swept. The most uniform arrangement at $P/Q$ is a Beatty sequence — particle $n$ at the floor of $nQ/P$ — and its gaps take exactly two values in the order a Sturmian word puts them, which is the rational end of the same construction that gives the Fibonacci chain at an irrational slope.

That makes the domain threshold a statement about reflections. A domain of $D$ sites broadens every reflection to a width of about $1/D$, and two locked states whose wavevectors differ by less than that are one blurred peak. The fractions with denominator at most $D$ are spaced at least $1/D^2$ apart, and those with larger denominators crowd between them; a domain of $D$ sites therefore resolves the locked states with denominator up to $D$ and smears the rest into what looks like a continuously moving satellite. This is the situation the extra dimension that makes it periodic describes from the other side, where a satellite that drifts with temperature is indexed with a second integer precisely because no small denominator fits it — and the situation two lattices in one crystal reaches when the two periods have no common multiple within any domain at all.

The two descriptions give the same answer, which is a check rather than a coincidence. A resolution in the chemical potential and a coherence length in the crystal are two ways of limiting the denominator, and both limits produce a Farey sequence because both act on the denominator alone.

Two limits on one axis #

The second threshold the earlier essay named is the coherent domain. A locked state at density $P/Q$ repeats every $Q$ sites, and a domain of $D$ sites holds fewer than one repeat when $Q$ exceeds $D$; a diffraction experiment on such a domain cannot distinguish that state from an incommensurate one, because nothing in the domain has repeated. So a step is observable only if its denominator is at most $D$ as well as its width at least $\delta$.

Both limits are bounds on the same integer, so they combine by taking the smaller. The visible steps are then all fractions with denominator up to $\min(Q^*(\delta), D)$, which is again a Farey sequence — the two thresholds do not produce a ragged set, only a shorter one.

Which limit binds is readable off the figure. For the long exponential at one part in a thousand the resolution allows denominators to twenty, and any domain larger than twenty sites changes nothing: the field is the limit. At one part in a million the resolution allows fifty-two, and a domain of twenty sites caps the count at 127 steps where the field alone would allow 829. For a long-ranged repulsion a better field buys little unless the crystal is also larger, and for a short-ranged one the reverse. That is the prediction the earlier essay asked for, in the only form a model without units can give it: not how many steps a particular crystal shows, but which of two improvements to an experiment would show more.

What the numbers depend on, and what they cannot say #

Nothing here is in kelvin or gigapascals. The chemical potential is in units of the repulsion, and the resolution is a share of the whole axis. Converting either to what an instrument reads needs the relation between the experiment’s controlling field and the model’s chemical potential, which belongs to a material and not to this model. The statements that survive the conversion are the structural ones: that the visible set is a Farey sequence, that the steps fill the axis, and how the visible denominator grows with resolution for a given shape of repulsion.

The model is one-dimensional, at zero temperature, with a postulated repulsion. Temperature widens nothing and narrows everything; at any temperature above zero the narrowest steps are washed out by entropy before any instrument reaches them, and that is a third threshold this page does not compute. A real modulated crystal also has a wavevector rather than a density and three dimensions rather than one.

The width law is quoted and checked, not proved here. Bak and Bruinsma derived both the closed form and the completeness of this staircase. What the computation adds is that both are checked against a staircase built from energies with no formula in it, fraction by fraction, and that the visible set is read off both routes and found to be a Farey sequence on each.

And no figure shows an incommensurate phase, because there is no interval of the axis to show one on. The strips draw steps and the spaces between drawable steps; that the spaces are themselves steps is established by the sum, not by the drawing, which at any resolution looks exactly as it would if the spaces were something else.

Why the count is a totient sum #

Nothing in the physics mentions coprimality. The totient enters because a step belongs to a density, and a density is a fraction in lowest terms: two sevenths and four fourteenths are one density and one step. So the number of steps with denominator $Q$ is the number of numerators coprime to $Q$, and the number with denominator at most $Q$ is the Farey count. The same fractions, ordered by size with each one’s neighbours satisfying $|PQ' - P'Q| = 1$, are what orders the gaps of a rotation and what the Fibonacci chain’s approximants are drawn from; a crystal that rounds an irrational off is choosing one of them.

What is new here is that the ordering by width and the ordering by denominator coincide exactly, so the Farey sequence is not merely the natural way to list the steps; it is the set an experiment of any resolution will see. The widest steps belong to the simplest fractions — the earlier essay found that as a tendency — and in this model the tendency has no exceptions at all, because width is a function of the denominator alone and a strictly decreasing one.

Still open: where temperature cuts the Farey sequence #

The one threshold left out is the thermal one. At a temperature above zero each step is a phase only if its window of chemical potential exceeds the free-energy cost of the defects that would carry the density across it, and that cost falls with temperature. The expectation is that temperature, too, cuts the staircase at a denominator — that the steps surviving at a given temperature are again every fraction up to some bound — because the defects that destroy a step at $P/Q$ are discommensurations whose energy depends on $Q$ in the same way the width does.

Whether that is exactly true or only roughly true is the question, and it is a calculation of a free energy rather than of an energy: a transfer matrix over the chain at finite temperature, which this model admits and nothing here runs. If it is exact, then every limit on seeing a staircase — field, domain and temperature — is a bound on one integer, and the visible phases of any experiment on such a chain are a Farey sequence without exception. If it is not, temperature is the first limit that produces a ragged set, and the order in which steps disappear on heating is a measurement of something the widths do not contain.