Statistical entropy

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Boltzmann's formula links entropy to microstates.

Key relationS = k ln Ω

S = k_B ln Ω, with Ω the number of microstates. More ways to arrange the particles means higher entropy.

Worked example

Gas expanding to double the volume: ΔS = Nk ln 2.

Common mistake

Microstates must be counted correctly (indistinguishability).

In more detail

Statistical entropy, from Boltzmann, is S = k ln W, where W is the number of microstates. It connects microscopic detail to thermodynamic entropy.

Statistical mechanics computes entropy of gases and solids from microscopic models.
Information theory uses similar entropy concepts.
Physicists use Boltzmann's formula to link the microscopic count of states to the measured entropy of a gas or solid.

What is a microstate?

A specific arrangement of positions and momenta of all particles.

Why does entropy increase?

Because systems overwhelmingly move to macrostates with more microstates.

What is the constant k in S = k ln Ω?

Boltzmann's constant, which relates the statistical count of states to the entropy in joules per kelvin.