Illustrated Physics
physics, drawn
Concept

Bifurcation — where it appears

A qualitative change in a system's long-term behaviour as a parameter passes a critical value. A fixed point loses its stability, or one periodic orbit is replaced by two, and the sequence of such changes is often universal — sharing exact numerical ratios with unrelated systems.

Named by 8 essays across 4 fields — each of them below, with the objects they name alongside it.

The reaction that reaches zero, and where the bead lets go. The force the sphere pushes back with, in units of the bead's weight, against the angle from the top. It starts at 1.000 and falls, because the speed the bead has gained needs more centripetal force than gravity's component along the radius can supply. At 48.19° it reaches zero, and past that the surface would have to pull inward to keep the bead on it — which a surface cannot do. So the bead leaves there, and the departure angle is a statement about the sign of a constraint force rather than about a speed or a height. The multiplier is what carries that sign: solve the motion in the angle alone and the reaction is absent from every equation, so nothing in the solution knows that the constraint has stopped holding, and the bead is drawn happily circling a sphere it has already left.

The force a coordinate cannot see

Writing a pendulum in terms of its angle is the first good move anybody learns, and it deletes the tension from every equation that follows. The string still breaks. Recovering the force that the clever choice of coordinate threw away turns out to be a computation with a sign in it, and the sign is where the bead leaves the sphere.

mechanics · Least action Every value the orbit of x → r x(1 − x) settles on. The values a long orbit of x → r x(1 − x) visits, one column of the picture for each of 320 settings of r between 2.8 and 4. A single point means the orbit settles to one value, two means it alternates, and each branching doubles the count with the gaps shrinking by a constant factor. The superstable settings marked run 3.23607, 3.49856, 3.55464, located by bisection on the map itself. They accumulate at r = 3.569946, and past it the orbit visits a band of values rather than a list of them. The bands are not noise: the map has no random number in it, and the same initial value gives the same orbit every time.

The map a dripping tap turns out to be

A universal result about one-dimensional maps is worth nothing to a physicist unless a real system is one, and a tap, a convection cell and a driven circuit are continuous systems with no map in sight. What makes them maps is dissipation — and the systems that have none take a different route entirely.

mechanics · Chaos Long stretches of order, broken without warning. 1800 successive values of the logistic map at r = 3.828427 − 0.00002, a distance of 2.0 × 10⁻⁵ below the setting at which its stable three-cycle is born. The shaded stretches are calm: the orbit repeats itself to within 0.004 every third step, cycling through three values as though the three-cycle already existed. Between them the orbit bursts through the whole interval with no discernible pattern, and then, at an unpredictable moment, is captured into another calm. In a run of 400,000 iterates at this setting the calms last 173 iterates on average, and the channel the orbit creeps through — measured on the map's own third iterate — has a gap of 4.1 × 10⁻⁵ and a longest passage of 253 iterates. Nothing random is added: the sequence is the same every time it is computed from the same start.

The calm that is the ghost of a cycle

Just before a chaotic system settles into a stable cycle it does something stranger than either: it behaves perfectly periodically for long stretches, and then, at moments nothing in the record predicts, bursts into disorder and back. The calm is a cycle that does not exist yet, creeping through the narrow gap where it is about to be born, and how long each calm lasts is set by the square root of the distance to that birth.

mechanics · Chaos Shake more gently and the grains choose a side. The steady share of the grains in each of two shaken compartments joined by a slot, against B, the parameter that grows as the shaking gets gentler. Below B = 4 the only steady state is an equal share, and it is stable. At B = 4 it becomes unstable and two clustered states appear, continuously, one with each compartment full: the fuller compartment holds 0.855 of the grains at B = 5, 0.929 at 6 and 0.979 at 8. Which compartment fills is decided by the first fluctuation. Nothing in the rules prefers either side, and the equal state that the rules would suggest is the one the grains avoid.

The sand that chooses a side

Divide a box of grains into two compartments with a slot in the wall between them and shake it from below. Shaken hard, the grains share themselves equally, as molecules of a gas would. Shaken a little more gently, they gather in one compartment and leave the other almost empty, and the crowded side is cold and the empty side hot. Nothing sorted them. Every collision between grains loses energy, and a compartment that happens to hold a few more grains loses more of it and stops throwing grains over the wall.

fluids · Granular matter The same paste stopping or running, according to the push. A thixotropic paste, sheared hard and then held under a constant stress: its viscosity against time, on a logarithmic axis, for stresses of 1.5, 1.8, 2.0, 2.4 Pa. Its structure builds at rest and is broken by flow, and the model's critical stress is 1.89 Pa. Below it the paste's viscosity climbs without limit: it slows, its structure builds faster than the slowing flow can break it, and it creeps to a stop, its viscosity growing in the end as the 1.5 power of time. Above it the viscosity falls: it speeds up, breaks its structure faster, and runs at a steady rate. After a thousand seconds the viscosities stand at 514.1, 238.3, 2.0, 1.5 Pa s: stresses a few per cent apart have gone opposite ways — a bifurcation, not a smooth change — and the paste hesitates longest when the stress is closest to the critical value.

The paste that forgets it was stirred

A jar of clay slurry left on a shelf sets like a jelly; shaken, it pours. That is thixotropy, and the tempting description — its viscosity falls when it is stirred — misses what makes it strange. A thixotropic paste has no viscosity at a given rate of stirring until it has been told how long it has been stirring, and under a steady push it cannot settle on a slow flow at all. Push slightly less than a critical stress and it creeps to a halt; push slightly more and it runs. Between the two there is nothing, and a paste made to flow slowly splits into a layer that flows and a layer that does not.

fluids · Rheology Five square logs of different densities, as they float. Cross-sections of five long square logs in their stable floating positions, with relative densities 0.1, 0.25, 0.5, 0.75 and 0.9, each drawn with its water line and its submerged part shaded, and a dot at its centre. The lightest floats flat and high; at 0.25 it has tilted to about 27°; at 0.5 it floats on its corner with a diagonal vertical and the water line across the middle; at 0.75 it tilts to the same angle as at 0.25, and at 0.9 it floats flat again, low in the water. The heavy logs are the light ones turned upside down, with air and water exchanged.

The log that floats on its corner

A long square beam of very light wood floats flat, one face up, as anyone would expect. A beam of wood half as dense as water does not: it rolls onto a corner and floats with a diagonal vertical, like a diamond. Between the two, at a quarter of water's density, it floats tilted at twenty-seven degrees, neither flat nor on edge. Heavier beams repeat the sequence in reverse, and the whole pattern is symmetric: a beam of density s floats at exactly the angle of one of density 1 − s. None of it depends on anything but a square, a density and Archimedes, and the same arithmetic decides which way an iceberg rolls when it breaks.

fluids · Buoyancy Two ways to spin, against angular momentum. The spin Ω²/πGρ of the Maclaurin spheroids (blue) and the Jacobi ellipsoids (red) against their angular momentum, in units of √(GM³R) with R the radius of a sphere of the same volume. The two families meet at the bifurcation, angular momentum 0.304. Below it only spheroids exist. Above it a body of given angular momentum could be either, and the two differ: the ellipsoid spins more slowly, its spin falling as its angular momentum grows, because it gets longer and its moment of inertia climbs faster than its angular momentum. Every Maclaurin spheroid past the bifurcation is a shape a slightly viscous body will leave.

The spinning fluid that becomes a cigar

A ball of fluid held together by its own gravity flattens when it spins, as the Earth bulges at the equator. Spin it faster and it flattens further, into a lens — but only up to a point, and long before that point something unexpected happens. In 1834 Jacobi found that a fast-spinning fluid can also take a shape that is not symmetric about its axis at all: an ellipsoid with three different axes, stretched along its equator, tumbling end over end. Past a definite spin the stretched shape has less energy than the flattened one, and a body that can shed energy will choose it.

astrophysics · Self-gravity The fold beyond which no heap is steady. The centre's steady temperature rise, in units of RTₐ²/E, against the Frank-Kamenetskii parameter δ, for a slab, a long cylinder and a sphere, found by integrating the steady heat equation outward from the centre: each shape has two steady states for every δ below a critical value — a cool, stable one and a hot, unstable one — and none above it. The fold is at δ = 0.878, 2.000, 3.322 for the slab, cylinder and sphere: a sphere tolerates nearly four times a slab's δ at the same half-width, because its surface is larger in proportion to its volume and heat from its centre has more ways out. A real heap, which is neither, has a critical δ between, and fixing it for a given shape is the first step in calculating the size at which a store becomes unsafe.

The heap that sets itself alight

A haystack, a coal stockpile, a bin of oily rags or a silo of damp grain can catch fire with no spark at all. A slow reaction inside makes a little heat; the heap holds it in; the warmer middle reacts faster and makes more. Whether that ends in a warm heap or a burning one is decided by a single number built from the heap's size, the air's temperature and the reaction's speed — and past one value of it there is no steady temperature at all. The critical size shrinks steeply as the weather warms, and the heap gives almost no warning before it goes.

thermodynamics · Diffusion

Named alongside it

The objects these essays reach for when they reach for this one.

AttractorChaosLyapunov exponentStabilityActivation energyAngular momentumArchimedes principleArrhenius lawBuoyancyCentre of buoyancyConstraintConstraint force

All concepts