A fluctuation is a temporary deviation from equilibrium, observed in quantum vacuum, thermal systems, or cosmological fields.Fluctuation is a spontaneous deviation of a macroscopic parameter (temperature, density, energy) from its equilibrium value. In closed systems, it is smaller the more particles there are, but in the micro-world, quantum fluctuations are fundamental: even absolute zero does not freeze the zero-point oscillations of fields.
The concept took hold in the 19th century: after Robert Brown's observations (1827) of the chaotic motion of pollen in water, Albert Einstein in 1905 quantitatively described Brownian motion, proving the existence of molecules. Ludwig Boltzmann had earlier laid the foundations for the statistical interpretation of thermodynamics, linking entropy to probability, and thus to fluctuations. In the quantum realm, the formalism was developed in the mid-20th century.
In thermodynamic systems, the probability of a fluctuation is exponentially suppressed, as prescribed by the Boltzmann distribution (P ~ e^{-ΔE/kT}). In quantum mechanics, even the ground state of a harmonic oscillator has non-zero energy, causing zero-point fluctuations of coordinate and momentum, which follows from Heisenberg's uncertainty principle. In cosmology, quantum fluctuations of the vacuum inflaton field, stretched to macroscopic scales, became the seeds of the large-scale structure of the universe, observed through anisotropies in the cosmic microwave background.
Without quantum fluctuations, galaxies would not have formed, and electronic circuits constantly suffer from thermal noise — voltage fluctuations due to random electron motion.
The cosmic microwave background radiation is an 'imprint' of quantum fluctuations of the early universe, magnified by inflation to sizes that gave birth to galaxies.
| R | variable | Flux ratio — flux ratio |
| Φ₁ | variable | First flux — radiant flux watt |
| Φ₂ | variable | Second flux — radiant flux watt |
| η_i(t) | variable | noise term at time t for component i — noise newton |
| η_j(t') | variable | noise term at time t' for component j — noise newton |
| T | variable | temperature — temperature kelvin |
| γ | variable | dissipation coefficient — dissipation coefficient kilogram per second |
| δ_ij | variable | Kronecker delta — kronecker delta |
| δ(t-t') | variable | Dirac delta function — dirac delta function per second |
| k_B | constant | relates energy to temperature — boltzmann constant = 1.380649e-23 joule per kelvin |
| ⟨...⟩ | operator | average over many realizations — ensemble average |
| C(\omega,\omega') | variable | Correlation function of fluctuations at two frequencies — correlation function |
| H(k;\omega) | variable | Transfer function or response kernel at wavevector k and frequency omega — transfer function |
| H^*(k;\omega') | variable | Complex conjugate of the transfer function at frequency omega' — transfer function |
| P(k) | variable | Power spectrum of the underlying fluctuations — power spectrum |
| \int dk | operator | Integration over all wavevectors — integral operator |
| ε | variable | ellipticity — ellipticity |
| I_xx | variable | moment of inertia about x-axis — moment of inertia kilogram metre squared |
| I_yy | variable | moment of inertia about y-axis — moment of inertia kilogram metre squared |
| I_zz | variable | moment of inertia about z-axis — moment of inertia kilogram metre squared |





















