Illustrated Physics
physics, drawn
Concept

Relativistic gas — where it appears

A gas whose particles move at nearly the speed of light, so that a particle of energy E carries momentum E/c rather than the square root of 2mE. Its pressure is a third of its energy density rather than two thirds, and its adiabatic index is 4/3, which is what limits the mass of a white dwarf and of a star.

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

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 Energy, pressure and entropy of a gas nobody counted. The energy density, pressure and entropy density of blackbody radiation against temperature, on logarithmic axes, together with the pressure a monatomic gas of the same energy density would have. Every curve is a power of the temperature — the fourth for energy and pressure, the third for entropy — because the only length in the problem is the thermal wavelength and the only energy is kT. The pressure is exactly a third of the energy density, where an ordinary gas's is two thirds, a factor of 2: a photon carries momentum E/c and a slow molecule carries √(2mE), and that difference is the whole of it. Some values: at room temperature the radiation pressure is 1.86e-6 pascals, which is a ten thousand millionth of an atmosphere; at 1e+7 kelvin it is 2.52e+12, which is where radiation rather than matter holds a star up.

The gas that nobody counted

A box of gas holds however many molecules were put in it. A box of radiation holds however many photons the temperature says, because the walls make and destroy them until the free energy is least — and one dropped assumption changes every result. The pressure becomes a third of the energy density instead of two thirds, the entropy goes as the cube of the temperature, and the adiabatic index comes out at exactly four thirds.

thermodynamics · Blackbody

Named alongside it

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

Adiabatic indexBlackbodyThe Boltzmann factorChemical potentialCosmic microwave backgroundDegrees of freedomEntropyEquation of stateEquilibriumEquipartitionGravitational collapseHeat capacity

All concepts