Illustrated Physics
physics, drawn
Concept

Brownian motion — where it appears

The ceaseless random movement of a small particle suspended in a fluid, driven by the unequal impacts of the molecules around it. Its mean square displacement grows in proportion to time with a coefficient containing Avogadro's number, which is how the number was first measured and how atoms stopped being a hypothesis.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

Seven walks from one point. 7 random walks of 400 steps each, all starting at the same place. None of them goes anywhere in particular and none of them stays put; the typical distance reached after n steps is the square root of n, so quadrupling the time doubles the spread. The tracks are seeded, so this is a property of the figure rather than of any one run.

The jiggle that proved atoms

A pollen grain in still water never stops moving. For eighty years that was a curiosity with no explanation; then it became the measurement that settled whether matter is made of particles, by turning a microscope and a stopwatch into a count of how many molecules are in a mole.

thermodynamics · Diffusion The fraction of walks that have come home, against how long they have walked. The proportion of 1,600 lattice random walks that have returned to their starting point at least once, against the number of steps taken, on a logarithmic horizontal axis, in one, two and three dimensions. In one dimension almost every walk is home almost at once and the fraction climbs toward one. In two it climbs more slowly — the return is still certain, but only logarithmically, so a two-dimensional walk that has not come back after a thousand steps is unremarkable. In three the curve flattens: it reaches 0.3481 and stops, against the exact value 0.3405, drawn as a line. That number is Watson's integral, and it is the probability that a three-dimensional walk ever comes home at all. The difference between the cases is not one of degree. In one and two dimensions the expected number of returns is infinite and a diffusing particle visits every site eventually; in three it is finite, and a molecule released in a room will, with probability two-thirds, never pass through its starting point again. The same statement runs the other way round: a reaction that needs two diffusing partners to meet is a very different problem on a membrane from what it is in a cell.

The walk that comes home

A particle wandering at random on a line returns to where it started, with certainty. On a plane it returns, with certainty. In three dimensions the probability is 0.3405 — so two out of three molecules released in a room never pass through their starting point again, and the difference between the cases is not a matter of degree.

thermodynamics · Diffusion A force that points across the beam, not along it. The intensity of a beam focused to a waist of 0.5 µm, and the force it exerts on a 60 nm polystyrene sphere in water, across the beam. The force is not a pressure and does not point along the light: it is the pull on an induced dipole sitting in a non-uniform field, proportional to the gradient of the intensity rather than to the intensity, and it points up the gradient — towards the bright axis from either side. It vanishes exactly on the axis, which is what makes the axis a trap rather than a place. The well is 47 times kT deep at 100 milliwatts, which is why the particle stays: thermal motion explores it and does not escape it. The stiffness near the bottom is 3.070 piconewtons per micrometre, and a particle in a spring that stiff wanders 36 nanometres from centre.

The light that pulls rather than pushes

Radiation pressure is momentum arriving, and it points along the beam. A gradient of intensity does something else entirely — it polarises a particle and then pulls the induced dipole up the gradient — so a focused beam holds a particle at its waist against the push, and the force that does it is not a pressure at all.

astrophysics · Radiation pressure A ratio in the exponent. The Boltzmann factor against the energy of a state measured in units of kT, with the logarithm up the axis so that the straight line is the whole content. Every kT of energy costs a factor of e, so 5, 12, 20 times kT are factors of 10^-2.2, 10^-5.2, 10^-8.7. At 295 K, kT is 25.4 meV, so a barrier of 0.4 eV is 15.7 kT and a factor of 1.5e-7. That is the sense in which a third of an electronvolt is not a small energy: it is small compared with a chemical bond and enormous compared with kT, and it is the second comparison that decides whether anything happens.

The temperature a molecule does not have

Temperature fixes a system's average energy and nothing more. The actual energy wanders, by an amount tied to the heat capacity, and the relative size of the wandering falls as one over the square root of the number of degrees of freedom — so a mole has a temperature and a molecule does not.

thermodynamics · Equipartition One temperature, four pawls, and nothing gained. The net rate of a ratchet whose gas and whose pawl are at the same temperature, against the load, for notches 2, 5, 10, 20 times the thermal energy deep. Every curve passes through zero at zero load and is negative everywhere else. The device is not merely unable to lift a weight; under any load at all it turns the wrong way and lets the weight down, converting its potential energy into heat in the gas. Making the notch deeper slows everything down — an exponential in the depth — and does not change the sign anywhere. That is the second law arriving as a mechanism rather than as a prohibition. Nothing was assumed about entropy; the pawl was simply allowed to be as warm as everything else, and its own fluctuations undo exactly the rectification it was there to provide. Any rectifier small enough for thermal noise to matter has this problem, and the rectifier being clever does not help, because the same noise reaches the cleverness.

The engine a fluctuation cannot run

A ratchet lets a shaft turn one way and not the other. Put a paddle in a gas on the same shaft and molecular collisions appear to become a lifted weight — an engine running on one reservoir. It does not work, and following exactly why turns the second law from a prohibition into a mechanism: the pawl is as warm as the gas, and it lifts whenever it is asked to.

thermodynamics · Heat engines Two pushes that add up to a friction. The force on a sodium-23 atom from each of two counter-propagating laser beams tuned 0.5 linewidths below resonance, at 0.1 of saturation each, and their sum, against the atom's velocity in units of the linewidth over the wavenumber. Each beam pushes along its own direction and is heard loudest by an atom moving towards it, because the Doppler shift brings the red-detuned light up into resonance. At rest the two pushes cancel exactly; moving, the atom scatters more from the beam ahead of it than from the one behind, and the difference points against the motion. Near zero velocity the sum is a straight line through the origin — a friction, with slope −0.0907 ħk² — and it is largest at 3.13 m/s, beyond which the atom has been Doppler-shifted out of resonance with both beams and the grip weakens. The damping time it implies for sodium-23's mass is 17.5 microseconds.

The friction made of light

Two laser beams pointed at each other push an atom both ways at once, and at rest the pushes cancel. Moving, the atom hears the beam ahead of it louder than the one behind, and the difference is a friction. The photons that supply the friction arrive one at a time, so they also kick — and the temperature where the two balance contains the width of a spectral line and nothing else.

astrophysics · Radiation pressure

Named alongside it

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

EquipartitionThe Boltzmann factorDiffusionFluctuation dissipationMean square displacementRadiation pressureRandom walkScattering forceThermal equilibriumAvogadro numberCalibrationThe Carnot cycle

All concepts