Illustrated Physics
physics, drawn
Concept

The Boltzmann factor — where it appears

The factor by which the probability of a state falls with its energy, whose exponent is that energy measured in units of kT. Because it is an exponential, an energy a few times kT is essentially forbidden and one a fraction of kT is essentially free — a distinction with no middle ground.

Named by 16 essays across 4 fields — each of them below, with the objects they name alongside it.

The isothermal atmosphere against the real one. Pressure as a fraction of its sea-level value, against altitude. The curves are the isothermal barometric formula at 220, 288, 400 kelvin, whose scale heights are 6.4, 8.4, 11.7 kilometres. The points are the measured standard atmosphere. At 20 kilometres the 288 kelvin model is 71 per cent out, because the air up there is not at 288 kelvin.

Why the air thins with height, and why that is the same law as the speeds

The pressure of the atmosphere falls exponentially with altitude, and the distribution of molecular speeds falls exponentially with energy. These are not two results that happen to look alike. They are one statement read on two axes.

thermodynamics · Kinetic theory One well, two wells, and the band they become. The energy levels of a chain of identical wells, for 1, 2, 3, 6, 12, 40 of them, with an on-site energy of -4 eV and a coupling of -0.9 eV between neighbours. One well has one level. Two split it into two, 1.80 eV apart. By 40 the levels have filled a band 3.59 eV wide, which is closing on the limit of four times the coupling, 3.60 eV — and no further widening happens however many more wells are added. The count of levels grows with the number of wells; the width of the band does not.

What happens when the wells get close

Two atoms brought together split one level into two. A thousand split it into a thousand, packed into a band whose width stops growing after the third. Whether that band is full or half full is the whole difference between a wire and a window.

quantum · Bands 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 10, 20, 30 times kT are factors of 10^-4.3, 10^-8.7, 10^-13.0. At 300 K, kT is 25.9 meV, so a barrier of 0.35 eV is 13.5 kT and a factor of 1.3e-6. 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 exponential that decides everything

Maximising the number of ways a reservoir can arrange what is left after taking E out of it gives one factor, e to the minus E over kT. Its exponent is a ratio, which is why a barrier of a third of an electronvolt — nothing at all by chemical standards — is the difference between instantly and never.

thermodynamics · Entropy Equipotentials, with the field lines that cross them. Contours of constant potential, traced by marching squares, with field lines traced along the gradient of the same potential. The two families meet at right angles everywhere, which is a consequence of the field being the gradient rather than a property of the drawing.

The attraction that needs no charge

Gauss's law says nothing comes out of a neutral molecule, and yet water's field one nanometre away reaches 1.1 × 10⁸ V/m. What survives when the monopole vanishes is a separation, and every step down the tower of falloffs below it is paid for with one more order of cancellation.

electromagnetism · The field concept Ways to arrange 12 coins. The number of distinct arrangements giving each number of heads, for 12 coins. Every individual arrangement is equally likely; the middle wins because there are more ways to reach it.

What a system actually minimises

A ball falls to the bottom of a bowl and a gas fills a room, and neither of those is the rule. A system in contact with a large reservoir minimises U − TS, and the minus sign is the reservoir's own entropy written in the system's variables — which is why a rubber band pulls harder when it is heated.

thermodynamics · Entropy The phase boundary of water, from one equation. Pressure against temperature for water on a logarithmic pressure axis spanning 6.6 decades. The vaporisation curve is integrated from Clausius–Clapeyron between the triple point at 273.16 kelvin and 611.7 Pa and the critical point at 647.096 kelvin, with a single latent heat of 43.32 kilojoules per mole — the value the two published points on the curve imply. The measured latent heats are 45.05 at the triple point and 40.65 at the reference point, and the fitted value sits between them, because a constant latent heat is an average over the interval. The sublimation curve below the triple point is not measured but predicted, from the two latent heats adding where all three boundaries meet: 51.1 kilojoules per mole, which reaches 103.2 Pa at 253.1 kelvin against a measured 253.15. The melting curve is drawn at the slope Clapeyron gives it, -13.5 megapascals per kelvin, which is a volume ratio and nothing else: water's solid is 917 against 1000 kilograms per cubic metre for its liquid, so melting shrinks it and the line leans backwards. Across the whole 6.6 decades of this axis that line moves 5.2 kelvin, and one atmosphere shifts the melting point by 0.0075 kelvin. At 1 atmosphere the boundary is crossed at 373.1 kelvin, where water boils. The one place the curve fails is its top end: a constant latent heat reaches 37.4 MPa at the critical temperature where the measured critical pressure is 22.1 MPa, 70 per cent high, because the latent heat falls to zero at the critical point and this curve does not know that.

A boiling point is a pressure, not a temperature

Nothing about water names one hundred degrees; the air does. Where a liquid turns to vapour in its bulk is fixed by what pushes on it, which makes the familiar figure a coordinate on a curve — 71 °C on Everest, 119.5 °C in a sealed pot — and one latent heat draws the whole curve.

thermodynamics · Phase change The average position of something that is only shaking. The mean displacement of an oscillator against temperature, taken as a Boltzmann average over each well rather than from any expansion of it, with the temperature measured against each well's own depth so that unlike bonds can share an axis. A symmetric well gives exactly zero at every temperature — heating a harmonic solid makes it vibrate harder and does not make it larger. The others drift outward, because the outward side is the shallower one, and the measured slopes are a pendulum 0.000, a chemical bond 0.766, a pair of atoms 0.150. Thermal expansion is not a property a spring has; it is one a spring lacks.

Why heating a perfect spring changes nothing

A harmonic solid vibrates harder when heated and does not get any longer. Thermal expansion lives entirely in the term that the harmonic approximation throws away — and so does the fact that a solid conducts heat at a finite rate, which is the same discarded term doing a second job nobody would have connected to the first.

thermodynamics · Harmonic approximation The field a classical partition function cannot see. The momentum plane of one classical charge, in units of the root-mean-square thermal momentum. With no field the Boltzmann weight is a set of circles about the origin; with a field the same circles sit about p = qA, displaced and otherwise unaltered, because the energy depends on p only through p − qA. Integrating over the whole plane therefore cannot notice the displacement, and the numbers beside the drawing are that integral evaluated at 6 displacements: the largest departure from the zero-field value is 3.3e-16, which is the precision of the arithmetic and not a physical effect. The classical free energy has no B in it, so the classical magnetisation is exactly zero at every field and every temperature — no paramagnetism, no diamagnetism, no ferromagnetism. Every magnetic material is therefore evidence of something classical mechanics does not contain.

The magnetism classical physics forbids

Write down the partition function of any collection of classical charges in a magnetic field, and the field cancels out. Not approximately, not to leading order — the integral is over all of momentum space and the field only shifts where the middle of it is. So classical statistical mechanics predicts no paramagnetism, no diamagnetism and no ferromagnetism, and a compass needle is a quantum instrument.

electromagnetism · Magnetisation What a coordinate is worth, by the shape of its energy. The mean energy stored in one coordinate, in units of kT, against the power with which that coordinate enters the energy. The curve is 1/n and the points are the same average obtained by integrating the Boltzmann weight numerically, agreeing to 3.3e-7 per cent at worst. an energy going as |x|^1 holds 1.0000 kT; an energy going as |x|^1.5 holds 0.6667 kT; an energy going as |x|^2 holds 0.5000 kT; an energy going as |x|^3 holds 0.3333 kT; an energy going as |x|^4 holds 0.2500 kT; an energy going as |x|^6 holds 0.1667 kT. The familiar half a kT is the case n = 2 and nothing more general than that: a coordinate whose energy is linear in it — the momentum of an ultrarelativistic particle, or a field with no restoring force but a constant tension — carries a whole kT, and one confined by a very steep wall carries almost nothing. Equipartition is a theorem about quadratic terms, and calling it a theorem about degrees of freedom is the substitution that makes it fail.

The share that is not half a kT

Equipartition is quoted as half a kT for every degree of freedom, and it is nothing of the kind. It is half a kT for every *quadratic* term. A coordinate whose energy is linear in it carries a whole kT, and a gas hot enough that its particles' energy is pc rather than p²/2m therefore holds twice what the counting says — which drops its ratio of specific heats to four thirds and puts a star on the edge of being able to hold itself up.

thermodynamics · Equipartition The charge a plasma hides. The potential around a charge in a plasma, as a fraction of the bare Coulomb potential at the same distance, against distance measured in screening lengths. The electrons crowd toward the charge and the ions move away until the rearrangement cancels the field, and what is left falls as exp(−r/λ_D) on top of the ordinary 1/r. At one screening length the potential is already down to 37 per cent of the bare value, at three to 5 per cent, and at ten to 4 × 10⁻⁵ — so a charge in a plasma is invisible beyond a few λ_D, and the long range of the Coulomb force, which is what makes electrostatics awkward everywhere else, is simply gone. The screening length at 10¹¹ m⁻³ is 3.8 mm at 300 K, 6.9 mm at 1000 K, 21.8 mm at 10000 K, rising as the square root of the temperature because a hotter electron is harder to hold in place. Drawn this way the three curves coincide exactly: the shape is universal and the only thing a plasma's density and temperature decide is the length written on the axis.

The long-range force that does not reach

The Coulomb force falls off as slowly as gravity does, which is what makes electrostatics awkward — every charge is in principle in contact with every other. Put the same charge into a plasma and it becomes invisible beyond a few millimetres, because the mobile charges around it rearrange until its field is cancelled. What is left is a screened potential with a definite range, and that range is what makes a plasma a plasma.

astrophysics · Plasma oscillation The barrier a new phase has to climb. The free energy of a droplet against its radius, at four supersaturations. Two terms compete: the volume term is a gain and goes as r³, the surface term is a cost and goes as r². At small radius the surface wins, so a droplet that forms by chance is more expensive than the vapour it came from and evaporates again; past a critical radius the volume wins and the droplet grows without limit. The maximum between them is the barrier. At S = 1.5 the critical radius is 2.66 nm and the barrier 533.8 kT, S = 2 the critical radius is 1.56 nm and the barrier 182.6 kT, S = 3 the critical radius is 0.98 nm and the barrier 72.7 kT, S = 5 the critical radius is 0.67 nm and the barrier 33.9 kT. The critical radius contains a few hundred molecules at low supersaturation and a handful at high, which is the first sign that a theory built on a surface tension and a bulk free energy is being applied outside its comfort. Both the critical radius and the barrier are located here by searching the drawn curve and checked against the closed forms 2γ/|Δg| and 16πγ³/3Δg², which agree to a part in a thousand.

The barrier a new phase has to climb

Water vapour three times supersaturated is thermodynamically desperate to condense and will sit there indefinitely if it is clean enough. The obstacle is that a droplet has to start small, and a small droplet is nearly all surface — so the first nanometre of every phase transition costs energy rather than releasing it, and what decides whether anything happens is the height of that cost divided by kT.

thermodynamics · Phase change The temperature that runs off the top of the scale. On the left, the entropy of a collection of two-level systems against how many of them are in the upper level. It rises, reaches 0.69314 per spin — which is ln 2, and is where half of them are up — and then falls, because a system with every spin up is as orderly as one with every spin down. On the right, the slope of that curve, which is one over the temperature. Below the maximum it is positive and ordinary: adding energy adds entropy, and the temperature is what everybody expects. At the maximum it is zero, which means the temperature is infinite. Past it the slope is negative — adding energy now removes entropy — and the temperature is negative. Such a system is not cold. It is hotter than any positive temperature whatever: put it in contact with anything at all and energy flows out of it, because that raises the total entropy. The quantity that orders systems by which way heat flows is not the temperature but its reciprocal, which runs smoothly from large and positive through zero to negative, and it is the temperature that has the discontinuity. None of this is possible unless the energy has a ceiling, which is why it happens in a spin system and not in anything that can move: a gas has no upper bound on its kinetic energy, so its entropy never turns over and its temperature is never negative.

Hotter than any temperature there is

A system whose energy has a ceiling can be pushed past the point where adding energy adds entropy. Its temperature is then negative — and negative temperatures are not cold. They sit above every positive temperature on the only scale that decides which way heat flows, and a working laser is at one.

thermodynamics · Third law The bands bend by the amount the Fermi levels differed. A junction between 1e+17 cm⁻³ p-type and 1e+16 cm⁻³ n-type silicon at equilibrium, with the Fermi level flat by construction — that is what equilibrium means — and the two band edges carrying the whole of the 0.774 V drop. The bending happens over 331.8 nm, and it is not symmetric: the depletion reaches 301.6 nm into the lightly doped side and only 30.2 nm into the heavily doped one, because the same exposed charge is reached sooner where there is more of it. An electron in the n-side conduction band therefore faces an uphill barrier of 0.774 V to reach the p side, while an electron already on the p side rolls downhill without any barrier at all — which is the asymmetry the whole device is, and it is drawn here before any current has been mentioned.

One level, and the field that bends the bands

Two pieces of the same crystal doped differently have their Fermi levels at different heights. Joining them cannot leave both, because a difference in electrochemical potential is precisely what makes charge move — and everything a diode does is the accounting of what had to happen for that one difference to reach zero.

quantum · Bands 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 The pair potential, and the two things it does to a gas. The Lennard-Jones potential between two molecules, in units of its own depth and range, with the Mayer function it produces at 1, 3, 8 times the well depth in temperature. The virial coefficient is minus the integral of that function over volume, so the two parts of the potential contribute with opposite signs: the steep repulsive core makes the function minus one there, giving a positive contribution — molecules take up room — and the attractive well makes it positive, giving a negative one. At low temperature the attraction dominates and a gas is easier to compress than an ideal one; at high temperature the core dominates and it is harder. Between them is one temperature at which they cancel.

The first correction to the gas law

An ideal gas has no forces between its molecules. The first correction to what it does is computable from those forces alone — one integral over the pair potential — and its sign flips at a temperature where a real gas obeys the ideal law without being ideal at all.

thermodynamics · Kinetic theory 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

Named alongside it

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

TemperatureEntropyEquipartitionFree energyPartition functionBrownian motionDegrees of freedomEquilibriumHeat capacityIntermolecular forcesKinetic theoryLatent heat

All concepts