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.
What a learner can do afterwards
- 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
1 · Read
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.
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.
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.
Derivatives set means and swings; noise at rest equals response when driven.
2 · Watch
Take it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.