What is my research about?

Level 1Anyone

Heat something up and, sooner or later, it changes.

Ice melts. Steel softens. Silicon turns liquid at 1414 °C. Underneath every one of those changes sits the same picture: atoms held in place by their neighbours, shaken harder and harder until the arrangement gives way.

Nothing in the animation above is choreographed. Each particle is pushed away by the ones around it and kicked at random by the temperature. Order at the bottom of the slider, and flow at the top, are what comes out of those two rules — not something painted on top.

We want to predict what a material does in the large from what its atoms do in the small.

Level 2Some physics

Temperature is only half the story. Squeeze the same atoms, and they settle into a different arrangement entirely.

Two knobs now, and a map. Every point on the map is a temperature and a pressure; the colour tells you which arrangement wins there. Crystal B is open and roomy, the structure a material prefers when nothing is pressing on it. Crystal A is denser — squeeze hard enough and packing tightly beats bonding nicely. Push the temperature up from either one and the order dissolves into a liquid. The melting line tilts upward because a compressed solid is harder to melt.

Watch the atoms while you cross a line. The lattice does not fade — it rearranges.

Now the part that makes this a statistical problem. At a fixed temperature and pressure, a material does not sit in one arrangement of atoms. It visits an enormous number of them, each with its own probability, and everything you can actually measure — a density, a melting point, a heat capacity — is an average over that cloud. This is exactly where the difficulty lives:

Which regions of that unimaginably large space of possible atomic configurations actually matter at each temperature and pressure?

For a more detailed introduction to statistical thermodynamics, which underlies the theory of phase transitions, you can have a look at my short article series on Atomistic simulation theory (Classical statistical mechanics).

Level 3Configuration space

Stop thinking of atoms in a box. One arrangement of 54 atoms is a single point in a space of 108 coordinates.

You cannot draw 108 dimensions, so we do what the real work does: pick a couple of quantities that capture the structure and project onto them. Here they are the average number of neighbours an atom has, and how orderly the directions to those neighbours are. The open crystal has few neighbours; the dense one has many; both are highly orderly. A liquid has plenty of neighbours in no particular arrangement, so it falls to the bottom.

Two numbers cannot capture a 108-dimensional space, and they are not meant to. They are a window — enough to see that the arrangements we care about sit in small, isolated boxes.

Now try the obvious thing. Scatter the atoms at random and see what you get. Press the button a few times and watch where the grey dots pile up.

They never land in a crystal. Random arrangements do not sit near the interesting ones; they occupy a different part of the map entirely. This demonstrates how complicated these high-dimensional configuration spaces are.

The arrangements that matter occupy a vanishingly small corner of a space too large to search. Modern atomistic sampling methods aim to systematically expore the configuration space.

For a more detailed introduction to configuration spaces you can have a look at my short article series on Atomistic simulation theory (Potential energy surface).

Level 4Sampling methods

If you cannot search the space, shrink it. Then shrink it again — and never let go of what you have already found.

This is nested sampling. Scatter a set of walkers over the space. Find the worst one — the highest in energy — write it down, and throw it away. Then replace it with a copy of a survivor, nudged around at random under one rule: it must stay below the energy of the walker you just retired. Repeat.

The ceiling only ever falls, so the pale region — everything already ruled out — only ever grows. The walkers are squeezed, step by step, out of the disordered sea and into whichever basins they can still reach. Because you recorded every walker you retired, and you know exactly how much the allowed volume shrinks each time, the discarded pile is not waste: it is the quantity that gives you free energies, and from those, the phase diagram.

Now watch the last panel with exchange switched off. At the highest pressure the dense structure in the top right is the stable one — it is the deepest point on that surface — and the run never reaches it. Not because it is unlikely, but because squeezing raises the barrier around that basin faster than it deepens the basin itself, so the way in is walled off long before any walker starting from disorder gets close. The run settles confidently into the wrong phase and reports a wrong answer, with no sign that anything went missing.

The first panel, at low pressure, has no such trouble. Its walls are low and its walkers wander into the dense basin early. But at that pressure the dense structure is not stable, so it is not an answer — just a place its walkers happen to visit.

That is why there is a ladder rather than a pair. The pressure at which a run can reach the dense basin is not the pressure at which that basin is worth keeping, so no single partner can do both. Replica exchange moves configurations up the ladder one rung at a time: every iteration, neighbouring runs pick a walker each and swap them, but only if each would be legal in the other's run — below the other's ceiling. That one condition keeps every run statistically valid, and it is enough to carry a dense-basin configuration from the bottom of the ladder, where it is easy to find, to the top, where it is the right answer and unreachable.

For a more detailed introduction to nested sampling and its replica-exchange variant you can have a look at my short article series on Atomistic simulation theory (A primer on nested sampling, Replica-exchange nested sampling).

Level 5The research frontier

Everything so far assumed the energy of an arrangement is free. It is the most expensive thing in the calculation.

Level 4 spent about half a million energy evaluations to map one small system. Every one of those, done properly, is a quantum-mechanical calculation costing hours of supercomputer time. Half a million of them is centuries. That is the wall this whole field runs into.

The way through is to not compute them. Train a cheap model to predict the energy, and use it instead. But a model is only trustworthy where it has seen data — and you do not know which arrangements matter until you have already sampled them, which is the thing you needed the energies for.

The way out of that circle is to let the sampler choose. Press Run one round and watch the two panels together.

On the left is the surface the model believes at the pressure you pick. Solid where it has data, washed out where it is guessing. Note, that the enthalpy is H = U + PV, so they all share the same underlying potential energy model U. Each dot is a configuration that cost a real calculation. Notice what happens as rounds accumulate: the surface sharpens only along the path the sampler walks, and stays vague everywhere else — harmlessly. Hit Show the true surface to compare. The model never becomes globally correct, and it never needs to.

On the right, one column per run on the ladder and one row per round, coloured by the phase that run actually settled into. Grey means it settled into neither — its walkers are still strung out somewhere that is not a crystal, which is exactly what you see happening on the left in the early rounds. This panel reports nothing the sampling did not find, so the two halves of the figure can never tell you different stories.

Early rows are mostly grey and disordered. As calculations accumulate where they matter, the runs start landing in basins, the columns sort themselves into open on the left and dense on the right, and the boundary between them settles near the dashed line. It stays noisy: eight rungs can only locate a transition to within the spacing between them, and a single run on a half-learned surface is not a precise instrument.

That is the whole loop: sample with the model, let the sampler nominate what it is least sure about, compute those properly, retrain. The sampler is both the explorer and the thing deciding where the next expensive calculation goes.

And the reason it is worth the trouble is in the counter under the panels: Roughly one true calculation for every 10000 the sampler needed. In the real thing that ratio is closer to a million to one, which is the difference between a phase diagram you can compute and one you cannot.

For what this looks like when it is done properly — on silicon, germanium and titanium, converging in ten to fifteen rounds against density-functional theory — that is the npj Computational Materials paper.