Crystallography & Symmetry
patterns derived from their groups
Concept

Ground state — where it appears

The arrangement of lowest energy a model allows under stated conditions, such as a fixed density or a fixed chemical potential. In a chain with a convex repulsion it is the most uniform arrangement at each rational density, which is what makes the lock-in staircase computable.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Every rational holds a window, and there is nothing in between. The ground state density of a chain of particles with a convex repulsion, against the chemical potential that sets how many of them there are. Every density with denominator up to 24 is a flat step of positive width — 177 of them — and the steps with the simplest fractions are the widest: a half takes 19 per cent of the whole range on its own. The risers between them are not smooth stretches; they are where the densities with larger denominators sit, and a finer computation fills them with more steps. What is left after every rational has taken its window is the irrational densities, which are the genuinely incommensurate ground states and have no width at all.

Every fraction holds a window

Three essays here name the devil's staircase and none computes one. A chain of particles with any convex repulsion has a ground state at every rational density holding an interval of chemical potential to itself — 709 of them computed, the widest taking 19% of the axis and the narrowest two parts in a million million — and the incommensurate densities are what is left over.

aperiodic · Modulation How many locked phases an experiment of a given resolution can see. For three convex repulsions, the number of steps of the staircase wider than a given share of the chemical-potential axis, as that share is made smaller by factors of ten. A resolution of one part in a million shows 199 steps for the short exponential, 829 for the long one and 41 for the inverse sixth power. Every count is exactly the number of fractions with denominator up to some bound, because every step with one denominator has one width. The short exponential and the inverse power cross: the power law shows fewer steps at every practical resolution and more at one part in a hundred thousand million.

An experiment sees a Farey sequence

A devil's staircase has a step at every fraction, and no experiment sees them all. Which ones it sees is not a matter of luck: every step with the same denominator has the same width, so a finite resolution shows every fraction up to one denominator and nothing beyond it — a Farey sequence, whose length is a sum of Euler's totient. How that denominator grows with the resolution is the one thing the interaction decides, and what is left unseen is steps, never an incommensurate phase.

aperiodic · Modulation

Named alongside it

The objects these essays reach for when they reach for this one.

CommensurateConvexityEnumerationIncommensurateLock inLong-range orderModulationCut-and-projectQuasiperiodicResolution

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