Fluids can be described by what can change without mattering

For more than a century, the classic equations for fluids, the Navier-Stokes equations for how fluids move dating to the 1800s, defined a fluid by imagining what it was made of: a smooth, continuous substance that flows. After two decades spent reconstructing fluid theory from first principles, physicists now describe a fluid by what can change without mattering.

That is what a symmetry means. A symmetry is a change that does not matter. Turn a square a quarter-turn and the shape looks unchanged, but turn it by a smaller angle and the difference is obvious. A circle has more symmetry, because spinning it any amount leaves its appearance unchanged.

The method for turning symmetries into physics comes from Kenneth Wilson, a physicist at Cornell University who in the 1970s worked out how to build a simpler description of a system that keeps only what matters at large scales. The trick was to start from the symmetries at tiny scales and keep only the math that still matters at large scales.

The first step is to list the symmetries at the smallest scale you understand. In a magnet, for example, atoms sit in a grid, so shifting the whole magnet by one grid spacing restores its original pattern, and each atom is also free to point in any direction.

Those symmetries decide which mathematical pieces may appear in the description. Next, imagine pulling back to a wider view, where fine detail becomes too small to matter. Wilson's calculations show which pieces swell or fade as the view widens.

Almost every piece dwindles away, and what remains are the pieces that record large-scale patterns, leaving a concise description that no longer depends on most microscopic details. "This is the power of Wilson's understanding," said Michael Landry, a physicist at the Massachusetts Institute of Technology. "You can just skip to the answer."

Applied to fluids, that skip is the payoff. Start with the symmetries of how matter can move and slosh, look past the molecules to larger scales, and the familiar fluid equations appear as survivors. The new theory gives a reason for why the equations look as they do, and shows that they follow from symmetries.