--- title: "Fluctuations, Linear Response and the Fluctuation-Dissipation Relation" description: "At fixed T,V,N in the canonical ensemble, Var(E)=k_B T^2 C_V. At fixed T and density, extensive short-range systems with finite correlation length usually have relative energy fluctuations proportiona" 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: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Fluctuations, Linear Response and the Fluctuation-Dissipation Relation At fixed T,V,N in the canonical ensemble, Var(E)=k_B T^2 C_V. At fixed T and density, extensive short-range systems with finite correlation length usually have relative energy fluctuations proportional to 1/sqrt(N). The fluctuation-dissipation relation links equilibrium fluctuations to dissipative linear response through temperature and frequency factors. Resistor noise also depends on measurement bandwidth. 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 relative-fluctuation scaling for extensive systems with finite correlation length and describes its limits - States what the fluctuation-dissipation relation connects and names one measurable case ## Lesson: Noise is a measurement At fixed V,N in the canonical ensemble, Var(E)=k_B T^2 C_V. T is in kelvin. Boltzmann's constant k_B and heat capacity C_V, the temperature derivative of mean energy at fixed V,N, both have units energy/kelvin. Their product needs T^2 for energy-squared units. Variance is the mean squared deviation from mean energy. Its square root is the root-mean-square (RMS) energy fluctuation. Voltage is energy transferred per unit charge. For an ideal resistor, resistance R relates applied voltage to driven current by voltage=R times current. In thermal equilibrium, charge-carrier motion produces Johnson-Nyquist voltage fluctuations even with no battery attached or driven current. A measurement records these fluctuations; no audible sound is required. **Example.** At equilibrium T, the classical positive-frequency voltage-noise spectrum is S_V=4 k_B T R in V^2/Hz. Assume constant R and photon energies much smaller than k_B T in the band. An ideal open-circuit measurement draws no current. With unit gain in the band and zero outside, mean-square voltage = 4 k_B T R delta f. Bandwidth delta f is the band's width in Hz. To infer R, divide mean-square voltage by the known factors multiplying R. **Example.** At the same positive T, Var(E) is proportional to C_V. If sample B has four times sample A's C_V, B has four times the energy variance but only twice the root-mean-square fluctuation. Compare variance with variance, and fluctuation size with its square root. Linear response means an observable's change is proportional to a sufficiently weak perturbation. Dissipative response converts drive energy to heat. The fluctuation-dissipation relation links equilibrium correlations or spectra to that response through temperature and frequency factors. Johnson-Nyquist noise is a measurable case: T and R set the voltage-noise spectrum; the same R dissipates energy when driven. Noise and response are distinct; predictions require the relevant conditions. For extensive short-range systems at fixed T and density, away from critical regimes and with finite correlation length, mean energy and energy variance usually grow in proportion to N. The root-mean-square fluctuation grows as sqrt(N). Dividing it by an extensive, nonzero mean energy gives relative size proportional to 1/sqrt(N). Equal growth of variance and mean does not imply constant relative fluctuation size. The usual size scaling assumes correlations do not grow across the whole sample. Near a critical point, correlations can extend far and fluctuation scaling needs further analysis. Long-range interactions can also change the result. Smaller N often means larger relative fluctuations in the regular regime, but size alone does not justify the rule in every system. **Recap.** Fixed T,V,N: Var(E)=k_B T^2 C_V. In extensive short-range systems with finite correlations, relative RMS size usually falls as 1/sqrt(N). Noise relates to dissipative response through T and frequency. *Original lesson, questions and flashcards written for this learning dataset.* ## 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) - [The Boltzmann Factor and the Partition Function](https://lightmysky.com/learn/science/the-boltzmann-factor-and-the-partition-function-mt_u9MSoTgJkF) ## Optional connections - [Brownian motion and its defining properties](https://lightmysky.com/learn/mathematics/brownian-motion-and-its-defining-properties-mt_T5KtWIk07Z) ## Opens up - [Phase transitions, order parameters and broken symmetry](https://lightmysky.com/learn/science/phase-transitions-order-parameters-and-broken-symmetry-mt_3DQjfNJ-Ef)