Fluctuation

A fluctuation is a spontaneous, temporary departure of a physical quantity — energy, density, magnetisation, voltage — from its average value in a system at thermal equilibrium. Fluctuations are not measurement errors: the system genuinely differs from one instant to the next as it exchanges energy and particles with its surroundings. In the canonical ensemble the mean-square energy fluctuation is fixed by the heat capacity, ⟨(ΔE)²⟩ = k_BT²C_v.

Their defining feature is how they scale with system size. The relative size of a fluctuation falls off as 1/√N with the number of particles, so for a macroscopic sample of 10²³ molecules it is around one part in 10¹² — invisible, which is why bulk thermodynamics can treat its variables as sharp. For small systems — nanostructures, single molecules, biological motors — N is small and fluctuations become a dominant feature of the behaviour.

Fluctuations are intimately tied to a system's response to external forces through the fluctuation–dissipation theorem, and they grow without bound near a critical point, where they can become large enough to scatter light.

The statistical theory of fluctuations was developed by Einstein and Smoluchowski in the first decade of the twentieth century, in their work on Brownian motion and on critical opalescence, providing some of the strongest early evidence for the molecular constitution of matter.