# Entropy and the number of ways

Entropy measures, on a logarithmic scale, how probable a state is: the entropy difference is Boltzmann's constant times the natural logarithm of the probability W. Two independent systems multiply their probabilities and add their entropies, and only a logarithm does both. §5 of the light-quanta paper uses this rule, which it calls Boltzmann's principle.

Written for this edition, not translated from Einstein. Editorial review pending.

## Why is entropy the logarithm of a probability?

Four particles move independently in a box. The chance that all four happen to be in the left half at a given moment is ½ × ½ × ½ × ½ = 1/16. For a hundred particles it is (½)¹⁰⁰, about 8 × 10⁻³¹: possible, but never seen. A state is more probable when more of the ways the particles can be arranged produce it.

Entropy measures this probability on a logarithmic scale. Boltzmann's principle, as §5 of the light-quanta paper states it, makes the entropy of a system a function of the probability W of its state, and that function must be a logarithm:

$$
S - S_0 = k_B \ln W
$$

S minus S nought equals k B times the natural logarithm of W.

Why a logarithm: two independent systems have a combined probability W = W₁ × W₂, while their entropies add, S = S₁ + S₂. Only a logarithm turns the product into the sum, and that is the argument §5 gives. The constant $k_B$ is R/N, which is how the paper writes it, and the paper prints the natural logarithm as lg.

Apply it to the box. Squeezing n independent particles from a volume v₀ into a part v of it has probability W = (v/v₀)ⁿ, so the entropy changes by S − S₀ = n $k_B$ ln(v/v₀), a decrease, since v/v₀ is less than 1. §5 derives exactly this, and remarks that it needs no assumption about the law by which the molecules move.

The probability here is a frequency: the fraction of moments at which the state is found. §5 insists on this, and criticizes calculations in which the equally probable cases are simply postulated.

## Worked example: Squeezing a gas into half its volume

1. Take one gram-molecule of gas, n = N = 6.02 × 10²³ particles, and ask for all of them in the left half: v/v₀ = ½.
2. The probability is (½) to the power 6.02 × 10²³, far too small to write out. Its logarithm is N ln ½, about −4.2 × 10²³.
3. The entropy change is $k_B$ × N × ln ½ = R ln ½ = 8.314 × (−0.693), about −5.76 joules per kelvin.
4. Thermodynamics gives the same number for compressing an ideal gas to half its volume at fixed temperature, R ln ½. Counting ways and measuring heat agree.

§6 of the paper finds the same form for dilute radiation, with E/(βν) standing where the gas has Rn/N, and reads it as radiation behaving, in this respect, like independent quanta.

## Where this lesson stops

This lesson stops at entropy as the logarithm of a probability. What the comparison with radiation shows, and what it does not, is §6 of the light-quanta paper.

## This lesson builds on

- [Logarithms: turning products into sums](/foundations/logarithms/)
- [Probability and independence](/foundations/probability-independence/)
- [Entropy and temperature](/foundations/entropy-temperature/)