---
title: "Fluctuations, Linear Response and the Fluctuation-Dissipation Relation"
description: "Equilibrium quantities fluctuate by an amount fixed by a derivative of the same free energy that gives their averages, so heat capacity measures the size of energy fluctuations. The same noise sets ho"
canonical: https://lightmysky.com/learn/science/fluctuations-linear-response-and-the-fluctuation-dissipation-relation-mt_Ibsvdyx7uP
source: https://lightmysky.com/learn/science/fluctuations-linear-response-and-the-fluctuation-dissipation-relation-mt_Ibsvdyx7uP.md
retrieved: 2026-09-12
---

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# Fluctuations, Linear Response and the Fluctuation-Dissipation Relation

Equilibrium quantities fluctuate by an amount fixed by a derivative of the same free energy that gives their averages, so heat capacity measures the size of energy fluctuations. The same noise sets how strongly a system responds when it is pushed, which is why resistance and thermal noise are one measurement.

Subject: Science · Area: Thermal & Statistical Physics · Ages 22 to 24
Page: https://lightmysky.com/learn/science/fluctuations-linear-response-and-the-fluctuation-dissipation-relation-mt_Ibsvdyx7uP

## Ready when they can

- Relates the variance of energy in the canonical ensemble to heat capacity
- Explains why relative fluctuations shrink as system size grows
- States what the fluctuation-dissipation relation connects and names one measurable case

## Lesson: Noise is a measurement

Averages come from free-energy derivatives, and so do fluctuations. In the canonical ensemble the energy variance equals temperature squared times heat capacity. Big heat capacity means big energy swings: the same derivative sets the mean and the noise.

**Example.** A resistor at rest hisses with no battery attached: thermal agitation of the same electrons that carry current. Drive it and you read resistance. The fluctuation-dissipation relation says the rest noise and the driven response are one number seen two ways.

Relative fluctuations shrink with size. Variances grow with particle number N while averages grow with N too, so the ratio of swing to mean falls as one over root N. Nanosystems swing visibly; bulk matter looks sharp.

**Tip.** Use the relation both ways: equilibrium noise predicts how the system answers a push, and measured response predicts the rest noise. Given one side you can quote the other, which is why resistance and thermal noise are a single measurement.

**Recap.** Derivatives set means and swings; noise at rest equals response when driven.

## Practice

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

## Needs first

- [The Microcanonical and Canonical Ensembles](https://lightmysky.com/learn/science/the-microcanonical-and-canonical-ensembles-mt_NSCYLK-brt)
- [Brownian Motion and Its Defining Properties](https://lightmysky.com/learn/mathematics/brownian-motion-and-its-defining-properties-mt_T5KtWIk07Z)
- [The Boltzmann Factor and the Partition Function](https://lightmysky.com/learn/science/the-boltzmann-factor-and-the-partition-function-mt_u9MSoTgJkF)

## Opens up

- [Phase Transitions, Order Parameters and Broken Symmetry](https://lightmysky.com/learn/science/phase-transitions-order-parameters-and-broken-symmetry-mt_3DQjfNJ-Ef)