Why Does It Matter Which Way You Came? The System Remembers

A system without this property responds only to its present input, so the same input always produces the same output. A system with it responds to its history as well, so the output depends on the path taken to reach the current input. Plotting output against input while cycling the input up and down produces two different curves rather than one, enclosing a loop, and the area of that loop measures how much the behaviour depends on history. The name comes from a Greek word meaning lagging behind, which describes what the output does relative to the input.

The property appears across quite unrelated systems:

• •Magnetic materials, which retain magnetisation after the field is removed
• •Rubber and other polymers, which return along a different path when unloaded
• •Thermostats, which switch on and off at deliberately different temperatures
• •Ferroelectric materials, used for memory devices
• •Wetting, where a drop advances and retreats at different angles
• •Elastic materials loaded past their limit, which do not return

The magnetic case is the one the term was coined for and it explains what a permanent magnet is. Applying a field to iron aligns the small regions within it that are already magnetised, and removing the field does not restore the original disorder, because moving those region boundaries back requires energy to overcome obstacles in the material. Some alignment therefore remains, which is exactly what makes a permanent magnet permanent. A field applied in the opposite direction is required to remove it. Materials with a wide loop hold their magnetism strongly and suit permanent magnets, while materials with a narrow one lose it readily and suit transformer cores where the loop area is wasted energy on every cycle.

Every trip round the loop dissipates energy as heat, and the amount is exactly the area enclosed, which turns an abstract curve into an engineering cost. A transformer core is magnetised and demagnetised many times a second, so it traces the loop that often, and the energy lost heats the core and must be supplied from the input, which is why core materials are chosen for a narrow loop and why transformers are warm. Rubber flexed repeatedly heats up for the same reason, which is why a tyre run underinflated fails, since the extra flexing generates heat faster than it can escape.

Engineers frequently introduce the property rather than removing it, and knowing why explains several familiar behaviours. A thermostat that switched exactly at the target temperature would chatter on and off continuously as the temperature hovered, wearing the contacts and the boiler, so it is designed to switch on below the target and off above it, with a deliberate gap between the two. The same logic gives a mouse button its click, keeps a digital signal from oscillating on a noisy edge, and stops an automatic door from opening and closing repeatedly on a marginal detection. In each case the gap is added to make the system decisive rather than sensitive.

Where output depends on the path taken rather than only on the present input, cycling the input traces two curves enclosing a loop whose area measures the dependence on history. Iron keeps some alignment when a field is removed, which is what makes a permanent magnet. Thermostats and switches have the gap designed in deliberately, so they act decisively instead of chattering at the threshold.