The hotter start that cools first

Entropy is a count found the second law in the arithmetic of arrangements, and the second law, with a probability attached found how often it is broken by small systems. What a system actually minimises found that a system in contact with a bath heads for the minimum of its free energy, and the exponential that decides everything found that the rates at which it gets there are set by Boltzmann factors over the barriers in its way. The entropy that depends on how fast it was cooled, in another part of the collection, found glasses that never arrive at all.

All of those say where a system ends up and what governs its rates. None says which of two starting points arrives first, and there is a natural assumption that fills the gap: the one that starts nearer. A cup of warm water put in a freezer should freeze before a cup of hot water, which has further to cool. The claim that the opposite sometimes happens is old — Aristotle recorded a version, Francis Bacon and Descartes repeated it — and in 1969 a Tanzanian schoolboy, Erasto Mpemba, made it famous by insisting, against his teacher, that hot ice-cream mix froze faster than cold. This essay separates what is known about the claim for water, which is little and mostly negative, from a precise version of it that is established, computable and has been seen: in systems with a slow and a fast way to equilibrium, a hotter start can overtake a cooler one.

Two ways back #

Consider the simplest system with more than one timescale. A particle can sit in one of two wells. The left well has a single low-energy state and, higher up, a crowd of twenty excited states; the right well has three states, a little above the left well’s ground state. A high barrier separates the wells and a lower one separates the left well’s ground state from its excited states. Put in contact with a bath, the particle hops between states at rates set by the barriers and the bath’s temperature, with the forward and backward rates between any two states in the ratio of their Boltzmann weights, so that the bath’s equilibrium is where the hopping balances.

A distribution that is not the bath’s equilibrium relaxes towards it, and linear algebra says exactly how — the same decomposition into decaying modes that the equation that only runs forwards found for diffusion. The relaxation is a sum of modes, each decaying exponentially at its own rate, and here there are two: a fast mode, which redistributes probability inside the left well across its low barrier, and a slow mode, which moves probability between the wells across the high one. With the numbers used here the slow mode takes six hundred times longer than the fast. Any starting distribution is some amount of each mode, and after the fast mode has died away, whatever amount of the slow mode it contained is all that is left — decaying slowly, and dominating the late approach to equilibrium.

So the question of which start arrives first becomes a question of how much of the slow mode each start contains, and that has nothing to do, in general, with how far away each start is.

Why Newton’s law of cooling forbids it #

It is worth seeing why the familiar picture of cooling rules the effect out, because that shows exactly what has to be added. Newton’s law of cooling treats a body as a single temperature, falling towards the surroundings’ at a rate proportional to the difference. The difference then decays as one exponential, $e^{-t/\tau}$, whatever it started at, and two bodies started at different temperatures keep their order for ever: the hotter one is always hotter, by a margin that shrinks in proportion. A body described by one number has one mode, and one mode cannot be missed. The summer that reaches the cellar in December found fine ripples of temperature dying away much faster than broad ones, which is why the one-number picture usually works for a small, well-stirred body.

Anomalous relaxation therefore needs internal structure: a state described by more than one quantity, so that there is more than one direction in which it can be out of equilibrium, and relaxation rates that differ between the directions. The three-state model has two such directions, one fast and one slow. A real cup of water has many — its temperature is not uniform, its dissolved gases and surface evaporation have their own timescales, and it can supercool — which is part of why the question for water is so hard. The mode analysis does not say water shows the effect; it says what would have to be true of any system that did.

A share that falls and rises #

The systems in the race are each prepared in equilibrium at some starting temperature — a warm start and a hot start — and then dropped into a bath at a colder one.

The slow mode moves probability between the wells, so the amount of it a start contains is set by how much the left well’s share differs from its share in the bath. The figure plots that share against temperature, and its shape is the whole trick. Cold, the ground state dominates and the left well holds most of the probability. Warmer, the three states of the right well, only a little higher in energy, gain weight faster than the left well’s ground state keeps it, and the left well’s share falls. Hotter still, the twenty excited states of the left well, high in energy but numerous, begin to count, and the left well’s share rises again. The share has a minimum.

Because it has a minimum, a hot temperature exists at which the left well’s share is exactly what it is at the bath’s temperature — here 0.901 against a bath at 0.30. A system prepared at 0.901 has its probability split between the wells just as the bath will want it. Its ground state is underpopulated and its excited states overpopulated, which is far from equilibrium, but that is an imbalance inside the left well, which the fast mode corrects. A system prepared at 0.50, only a little warmer than the bath, has the wrong split between the wells, and only the slow mode can fix it.

How much of the slow way each start needs #

Computing the slow mode’s amplitude for every starting temperature makes it explicit. It is zero at the bath’s own temperature, where nothing needs to relax. It grows as the starting temperature rises — the warm start at 0.50 has a large amount — and then, instead of growing without limit, turns round and passes through zero again at 0.901, where the hot start’s wells already have the right shares. Beyond that it grows again, with the opposite sign.

The dependence of the slow mode’s amplitude on the starting temperature is non-monotonic, and that is the necessary and sufficient condition for the effect. Zhiyue Lu and Oren Raz set it out in these terms in 2017: in any system relaxing by a master equation, an anomalous relaxation — hotter overtaking cooler — occurs exactly when the coefficient of the slowest mode is a non-monotonic function of the starting temperature. A strong version, in which a hot start has zero slow-mode amplitude and relaxes exponentially faster than all its neighbours, occurs at the temperatures where the coefficient crosses zero.

The race #

The figure runs the race. The measure of distance is the relative entropy between each system’s distribution and the bath’s — the quantity what a system actually minimises found decreasing monotonically for any system relaxing in contact with a bath, equal to the excess free energy divided by the bath’s temperature. The hot system starts five times further away. Both fall quickly at first, as their left wells equilibrate internally. Then the warm system stalls: its fast mode is spent, and what remains of its distance is slow-mode amplitude, draining across the high barrier over thousands of time units. The hot system has no slow-mode amplitude to drain, and its distance keeps falling at the fast rate. It overtakes the warm system at about seven time units and is ahead by many orders of magnitude from then on.

The second law is untouched. Each system’s distance from equilibrium falls at every moment, exactly as it must. What the law does not say — and what intuition wrongly adds — is that two different starting points must keep their order. The distance from equilibrium is a single number, and the route back has many dimensions.

There is a subtlety in the word “temperature” that the race exposes. Once a system is dropped into the bath it has no temperature: its distribution is not a Boltzmann distribution at any temperature, and it will not be one again until it has fully relaxed. “Cooling faster” has to mean approaching the bath’s distribution faster by some measure of distance, and different measures can disagree about the ordering at intermediate times. The relative entropy is the natural one — it is the free energy that could still be extracted from the difference, divided by the bath’s temperature — and the crossing in the figure is robust against other reasonable choices at late times, because at late times every measure is dominated by the same slow mode, and the hot start has none of it.

Which starts arrive first #

Measured at a single late time, the distance from equilibrium as a function of starting temperature makes the ranking visible. Starts close to the bath’s temperature are nearest, as expected. The distance grows as the start gets warmer, up to the region where the left well’s share is at its minimum — and then, instead of continuing to grow, collapses. At the hot start’s temperature it is many orders of magnitude smaller than at any start near 0.50, because only the fast mode remains, and the fast mode has long since decayed. Every start whose slow-mode amplitude is smaller than a given warm start’s eventually overtakes it, however much hotter it began.

Two routes to one point #

The geometry is clearest in the space of probabilities itself. Every starting point lies on the curve of equilibrium distributions at different temperatures. Released into the bath, each distribution moves first along the fast direction — trading probability between the ground state and the excited states of the left well, leaving the right well’s share unchanged — and then creeps along the slow direction, adjusting the right well’s share, to the bath’s equilibrium. The warm start’s fast leg ends well away from the bath’s point, at the wrong right-well share, and the creep is long. The hot start’s fast leg ends at the bath’s point, because the curve of equilibria passes back through the bath’s right-well share at 0.901. A start hotter still overshoots and must creep back the other way.

That is the whole mechanism: the curve of equilibria, which a system’s starting state must lie on, happens to bend back and cross the line of states that the fast mode alone can reach from the bath’s point. Where it crosses, the slow route is not needed.

Seen in a laboratory #

The first clean demonstration of the strong effect came in 2020, from Avinash Kumar and John Bechhoefer. They held a single glass bead of 1.5 micrometres in water with optical tweezers, and by feeding back on its position built a tilted double-well potential for it, with the well depths and the barrier chosen by computer. The bead was then released from a distribution corresponding to a high “temperature”, enforced by random kicks, into the potential at the water’s temperature, and its distribution was measured over many repetitions. Hotter starts overtook cooler ones, and at a predicted starting temperature the relaxation was exponentially faster, exactly as the mode analysis says. Because every part of the system — its potential, its temperatures, its dynamics — was known and controllable, the observation left no room for the uncontrolled effects that plague experiments with water.

Similar anomalous relaxation has since been predicted or seen in spin systems, granular gases, quantum systems and clathrate hydrates, and the inverse effect — a cold start warming faster than a less cold one — has also been demonstrated. The general statement is not about heat at all: any system whose approach to equilibrium has well-separated rates can have starting points that miss its slowest mode.

And the water? #

For water itself the evidence is much weaker, and honesty requires saying so. Water freezing in a freezer involves cooling, evaporation, convection, the dissolving and outgassing of air, supercooling below 0 °C before ice nucleates, and heat transfer through the container to whatever it stands on — each of which depends on the starting temperature and on details of the setup. Hot water evaporates more, leaving less to freeze; it loses dissolved gases, which can change how far it supercools; it melts the frost beneath its container and improves its thermal contact. Any of these could make a hot sample freeze first in some conditions.

Whether they do reliably is another matter. The barrier a new phase has to climb found that the temperature at which supercooled water finally freezes varies from run to run, and the variation is large enough to swamp any small systematic difference in freezing time. A careful study published in 2016 by Henry Burridge and Paul Linden, which reviewed the literature and repeated the measurement with controlled conditions, found no evidence of the effect in the time for water to cool to 0 °C, and attributed many earlier positive results to the placement of the thermometer. A competition by the Royal Society of Chemistry in 2012 for the best explanation received thousands of entries and produced no agreed answer. What remains is a famous claim that may occur under particular conditions for reasons unconnected with the mode analysis, and a precise effect, with the same name, that occurs in simpler systems for a reason that can be written down.

Where the model stops #

The three-state model is a caricature chosen to make the mechanism visible: two wells, one barrier each, rates obeying detailed balance with Arrhenius factors, and a starting state in equilibrium at a sharp temperature. Real systems have many states and many modes, and the effect then depends on the slowest mode alone only after all the faster ones have decayed. The model’s temperatures are in units of its own energy scale, not kelvin, and its times in units of its own rates. It also assumes the system is coupled to a bath that stays at a fixed temperature — the cold reservoir absorbs whatever heat it is given without warming — whereas a cup of water in a freezer is a large system losing heat through several paths, whose own temperature is not even uniform, and which is far from the linear, weakly perturbed regime in which the mode expansion is exact.

What the pictures cannot show #

The figures show probabilities and their distances from equilibrium; they do not show what a thermometer would read. The system’s energy, which is what a thermometer tracks, is one particular average over the distribution, and it can relax with a different ordering from the relative entropy: two starts can cross in one measure and not in another. The figures also cannot show a single realisation. Each curve is the evolution of a probability distribution, the average over many copies of the system, and one particle hopping between wells looks like nothing but random jumps; the effect is visible only in the statistics of many.

Still open: when a large system has a slow mode it can miss #

The mode analysis is exact for systems described by a master equation, and it predicts anomalous relaxation whenever the slowest mode’s amplitude depends non-monotonically on the starting state. For systems with an enormous number of degrees of freedom — a liquid, a glass, a magnet near a phase transition — the modes form a near-continuum, the slowest ones are collective and hard to compute, and whether their amplitudes behave in the required way for realistic starting states is not generally known. Numerical studies of spin glasses and of model liquids have found Mpemba-like crossings; whether anything of the kind operates in freezing water, beneath the uncontrolled effects that dominate the experiments, is open, and it is a question that would need a measurement as clean as the bead’s to settle.

The habit worth carrying away is to ask how many routes a system has back to equilibrium before assuming that nearer means sooner. Relaxation is a sum of modes, and after the fast ones are spent only the slowest remains — so a start that happens to contain none of it, however far away, overtakes one that is nearer but needs it, and a three-state system started at 0.901 beats one started at 0.50 by orders of magnitude. The second law says every distance must fall; it does not say two distances must keep their order.