What is the glass transition, and why does cooling rate decide a sound?

The physics

A race between cooling and rearranging.

Most phase changes have a sharp, well-defined temperature: water freezes at 0°C, full stop, at a given pressure. The glass transition does not work that way. There is no fixed temperature at which a liquid "becomes" glass. Instead there is a competition between two timescales. One is how fast you are removing heat. The other is structural relaxation time, the time the material's own microscopic arrangement needs to explore its options and settle into whatever configuration is locally favored at the current temperature.

Near a material's melting point, relaxation is fast: molecules rearrange in nanoseconds, easily keeping pace with any cooling a person could apply, and the liquid crystallizes. As temperature falls, relaxation time grows, often by many orders of magnitude over a narrow temperature window, because rearranging requires cooperative motion of many neighbors at once and that motion becomes exponentially harder as the material densifies. At some point relaxation time exceeds the time available before the next temperature drop. The structure can no longer keep up. It gets left behind, and what was a supercooled liquid becomes, operationally, a solid with the disordered structure of a liquid: a glass. Keep cooling from there and the structure is essentially locked; there is no more rearranging to do in any practical amount of time.

The structural takeaway for an instrument: whether a system lands in the ordered state or the frozen disordered one depends on the race between the cooling clock and the relaxation clock, not on temperature alone. Cool the exact same substance to the exact same final temperature by two different clocks and you get two different structures. That is the entire thesis Anneal is built to make audible.

The equations

A temperature that remembers the path taken to reach it.

A. Q. Tool, working on optical glass at Corning in 1946, needed a way to describe glass that had been cooled at different rates without abandoning the language of equilibrium thermodynamics. His answer was fictive temperature, \(T_f\): the temperature at which the glass's current frozen-in structure would be the true equilibrium structure, if you could instantaneously re-equilibrate it there. A slowly cooled glass has a \(T_f\) close to its actual temperature, because it had time to track equilibrium most of the way down. A quenched glass has a \(T_f\) well above its actual temperature, because its structure froze in while it still "believed" it was warmer. \(T_f\) is not a temperature you can read off a thermometer. It is a bookkeeping variable that encodes trajectory, the specific path the cooling took, inside a single number.

This drift equation is the core of the Tool–Narayanaswamy–Moynihan model: Narayanaswamy formalized Tool's picture in 1971 with a rigorous relaxation-function treatment, and Moynihan and coworkers in 1976 fit it against real glass-forming systems and established the model as the standard one-order-parameter description of glassy relaxation. When \(T\) drops faster than \(\tau(T)\) allows \(T_f\) to follow, the gap between \(T\) and \(T_f\) widens and gets locked in as the material passes below its transition. When \(T\) drops slowly relative to \(\tau(T)\), \(T_f\) tracks \(T\) closely and the frozen structure ends up close to the true equilibrium one: a crystal-like outcome.

The relaxation time itself follows an Arrhenius law, steepening sharply as temperature falls:

In Anneal's reduced units, this law places the glass transition temperature \(T_g\), the point where \(\tau = 1\) second, at \(T \approx 0.535\), about 45% up the TEMP knob's throw. Below that, structural rearrangement takes longer than a second and the lattice is effectively frozen on musical timescales. Above it, rearrangement is fast enough to track the cooling in real time.

The measurement

Frozen disorder falls linearly against log cooling time.

The practical consequence of an Arrhenius relaxation time is a specific, measurable relationship: the amount of disorder that survives the cool, and the fictive temperature it corresponds to, both fall roughly linearly as a function of the logarithm of the cooling time. Doubling the cool time does not halve the frozen disorder; it takes a full order of magnitude in time to meaningfully move the outcome, because the relaxation clock itself moves exponentially with temperature. Anneal's RATE knob sweeps cool time from 0.03 to 20 seconds and the lattice traces this law directly:

Read across that table and the shape is unmistakable: roughly every threefold change in cooling time produces an audibly distinct amount of frozen disorder, with diminishing returns as the cool approaches the slow end, exactly as a logarithmic law predicts. A 30-millisecond cool locks in nearly 20 times the spread of a 20-second cool, at the identical final TEMP setting.

From law to instrument

A lattice that either settles or freezes.

Anneal's engine is a 32-site lattice, four microdomains per site, each carrying a per-mode configurational deviation \(d\) measured in cents of detune. The deviation at every site evolves under an exact Ornstein–Uhlenbeck update, run in float64 so the recursion is unconditionally stable at any cooling rate:

The equilibrium target itself only opens up once the lattice is hot enough to melt in the first place:

\(C_{\mathrm{DIS}}\) is set by the STRESS knob, the ceiling on how much disorder the molten state can hold. The sigmoid, centered at \(T_m = 0.80\) with a narrow 0.05 width, is what gives Anneal a real melting point rather than a gradual softening: below \(T_m\) the equilibrium target collapses toward zero and there is nothing left to relax into, so the lattice can only cool toward order. Above \(T_m\) the material is genuinely liquid, free to explore disorder up to the STRESS ceiling.

Everything downstream follows from that one recursion running independently at each of the 32 sites. TEMP sets the instantaneous \(T\) that drives both \(\tau(T)\) and \(\mathrm{var}_{\mathrm{eq}}(T)\). RATE sets how fast \(T\) itself is allowed to move when MELT is released, which is the direct control on the cooling clock discussed above. Each site's frozen mean deviation detunes its partial, so a crystal state (small, converged deviations) sounds coherent and in tune with itself, while a glass state (large, scattered deviations) sounds detuned and beating. JITTER is deliberately kept outside this recursion: it is a fast thermal vibration that always tracks the current \(T\), never the frozen \(T_f\), so it never gets locked in. It is the one part of the sound that stays alive even in a fully frozen glass, and because left and right channels sample the lattice independently under JITTER, it is also the plugin's sole driver of stereo width.

The plugin's own display, the Slab, draws all 128 microdomain deviations as dots around a centerline, with violet dashes marking each site's frozen mean and a telemetry line reporting \(T_f\), a GLASS or CRYSTAL word classification, spread in cents, and beat rate in Hz. Hold MELT and the dots visibly boil upward as \(T\) rises past \(T_m\); release it and they fall back toward the line at whatever rate RATE specifies. Slow enough, they land in a tight crystal band. Fast enough, they freeze mid-flight as glass. See note 10 for what that path-dependence proves quantitatively, including an adversarial control built specifically to fail at it.

Interactive lab

Choose a cooling time. Watch it freeze.

The left panel shows a temperature ladder on the vertical axis, with the glass transition \(T_g\) and melt point \(T_m\) marked, and roughly a hundred oscillator dots clustered around a centerline. Press Melt & release and the dots rise to the molten band, scatter with equilibrium disorder, then descend as \(T\) ramps down over your chosen cooling time. Slower cools give the lattice time to track equilibrium at each falling temperature, so the dots converge tightly onto the centerline: a crystal. Faster cools yank \(T\) down before the dots can follow, and they lock into whatever spread they happened to have when they crossed \(T_g\): a glass. The right panel accumulates one point per run, plotting frozen spread in cents against the logarithm of the cooling time you used, tracing the rate law from the previous section as you experiment. The readouts on load are pinned to Anneal's own documented 8-second-cool result (0.95 cents, \(T_f\) 0.526, CRYSTAL) before you run anything, so the starting numbers on this page match the product's own measured behavior exactly.

Honesty

What Anneal does and does not simulate.

The thesis is structural isomorphism, not science as decoration, and that demands precision about the claim. Anneal implements the Tool–Narayanaswamy–Moynihan model at the level real glass physicists use it for a single order parameter: an Arrhenius relaxation time, a fictive-temperature drift equation, and an exact Ornstein–Uhlenbeck update for the disorder itself. None of that is a metaphor. The recursion is the same one written in glass-science papers, run per lattice site, in double precision, unconditionally stable regardless of how fast you sweep RATE.

What is a design choice, not a physical derivation: the decision to express configurational deviation in cents of detune rather than a generic dimensionless order parameter, so the disorder is audible as pitch spread rather than requiring translation. The reduced temperature units, where \(T_g \approx 0.535\) and \(T_m = 0.80\) sit at specific fractions of a 0–100% knob throw, are chosen for a musically useful range on a MIDI controller, not measured from any particular real glass. And the sigmoid width \(W = 0.05\) that gives the melt point its sharpness is a deliberate design decision about how abrupt melting should feel under the hand, not a fit to a specific material's heat capacity curve.

That boundary is the point, not a hedge. A mapping earns its physics by producing a discernible, lawful difference in the sound, and the test is whether changing RATE changes the frozen spread the way the exponential relaxation law says it should. It does, by close to a twenty-to-one ratio across the knob's range. The mechanism transfers; the specific material constants were never claimed to.

References

Source material.

Related

Continue the mechanism trail.