---
title: "Microstates, Multiplicity and Boltzmann's Entropy"
description: "Counting the arrangements a system can take gives a multiplicity, and its logarithm times Boltzmann's constant is the entropy. The second law becomes a statement about which outcomes have overwhelming"
canonical: https://lightmysky.com/learn/science/microstates-multiplicity-and-boltzmanns-entropy-mt_55I3VvYMc6
source: https://lightmysky.com/learn/science/microstates-multiplicity-and-boltzmanns-entropy-mt_55I3VvYMc6.md
retrieved: 2026-09-12
---

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# Microstates, Multiplicity and Boltzmann's Entropy

Counting the arrangements a system can take gives a multiplicity, and its logarithm times Boltzmann's constant is the entropy. The second law becomes a statement about which outcomes have overwhelmingly more arrangements.

Subject: Science · Area: Thermal & Statistical Physics · Ages 20 to 21
Page: https://lightmysky.com/learn/science/microstates-multiplicity-and-boltzmanns-entropy-mt_55I3VvYMc6

## Ready when they can

- Counts microstates for a small system and identifies the most probable macrostate
- Relates entropy to multiplicity with Boltzmann's formula
- Explains why the second law is about probability rather than prohibition

## Lesson: Counting arrangements: multiplicity and entropy

Zoom in far enough and thermodynamics becomes counting. A microstate is one complete microscopic snapshot: every molecule pinned to a position with a velocity. A macrostate is the blurred version you observe: just pressure, volume, and temperature. Multiplicity, written W, counts how many microstates hide inside one macrostate, and the counts are wildly uneven. Toss four coins: all heads happens one way, but two heads and two tails happens six ways, so the mixed outcome dominates.

**Example.** Boltzmann's formula turns the counting into entropy: S equals k times the natural log of W. The logarithm matters because it keeps entropy additive. Put two independent systems side by side and their multiplicities multiply, while the log turns that product into a tidy sum. Twice the system, twice the entropy, exactly as a state function demands. Ink spreads through water because the mixed macrostate owns enormously more microstates than the separated one.

The second law survives, but its character changes. It no longer forbids entropy decrease the way energy conservation forbids free work. Instead it reports that decreases are fantastically improbable for large systems. A gas compressing itself into a corner is possible, yet you would wait past the end of the universe for it. Small systems are the exception: with a handful of particles, entropy can flicker downward briefly, exactly as the fluctuation theorems quantify.

**Tip.** Solve counting questions with one routine. List every microstate, group them into macrostates, and crown the biggest pile: that is equilibrium. Take logs to report entropy, and remember the scaling: squaring the multiplicity doubles the entropy. Never call the biggest pile forbidden to leave; it is simply where the system spends its time.

**Recap.** Entropy counts arrangements through S equals k ln W, the biggest pile of microstates is equilibrium, and the second law is overwhelming probability.

## Practice

8 questions on this page, each with its working shown.

## Needs first

- [Entropy as a State Function](https://lightmysky.com/learn/science/entropy-as-a-state-function-mt_bg_K5nU-GZ)
- [Discrete Random Variables and Probability Distributions](https://lightmysky.com/learn/mathematics/discrete-random-variables-and-probability-distributions-mt_yKztNkvvq3)

## Opens up

- [The Boltzmann Factor and the Partition Function](https://lightmysky.com/learn/science/the-boltzmann-factor-and-the-partition-function-mt_u9MSoTgJkF)
- [The Ising Model and the Mean-Field Approximation](https://lightmysky.com/learn/science/the-ising-model-and-the-mean-field-approximation-mt_wPwyBTVI0C)