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The most crucial accomplishment of Lev Landau

by Cyril Ulyanov

04-06-2026

Scientists put liquid helium into a container and cooled it down until it was freezing cold. Suddenly, they noticed liquid's behavior was strange: it became a superfluid with zero friction. First of all, it somehow oozed out through microscopic holes in the glass. Even more surprisingly, it crawled up the glass and flowed from the brim. According to regular physics, atoms are supposed to collide, create friction, and therefore decelerate. However, what they observed with the liquid looked like a contradiction.

Scientists tried to explain the lack of friction by looking at the liquid atom by atom, treating it like a very cold gas. They thought that when the liquid got ultra-cold, individual helium atoms changed their properties, which resulted in what they witnessed.

Lev Landau took a completely different approach. Instead of looking at individual atoms, he studied the liquid as a whole.

When liquid helium is cooled below a specific temperature, it stops acting like a single substance. Mathematically, it splits into two completely different liquids occupying the very same space. Scientists express this using the formula:

ρ=ρn+ρs \text{ρ} = \text{ρ}_{\text{n}} + \text{ρ}_{\text{s}}

The variables are respectively total density, normal fluid density, and superfluid density.

vc=minp(E(p)p) \mathrm{v}_{\mathrm{c}} = \mathrm{min}_{\mathrm{p}} \left( \frac{\mathrm{E}(\mathrm{p})}{\mathrm{p}} \right)

The formula calculates the speed limit (vc)(\mathrm{v_c}) the liquid must stay under so it doesn't have enough energy (E(p))(\mathrm{E(p)}) and momentum (p)(\mathrm{p}) to trigger friction. In other words, below this speed limit, the liquid lacks the energy to cause friction, so it moves freely.

To find this speed limit, scientists look at how energy moves through the liquid in two different ways: as phonons (sound waves) and rotons (microscopic twists). Moving fast enough to create a sound wave requires a high velocity. However, because helium atoms are closely packed together, they reach a threshold where the liquid is guaranteed to generate a roton twist first. Because Landau's formula calculates the minimum speed that makes friction, this lower threshold sets the speed limit to a lower value. The moment an object moving through the liquid exceeds this limit, it triggers these internal vibrations, which generates friction.

Landau published these formulas in the 1940s and earned the Nobel Prize in Physics in 1962.


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