concept

fluctuation

A fluctuation is a temporary deviation from equilibrium, observed in quantum vacuum, thermal systems, or cosmological fields.
22 related articles · 4 formulas · 1 who worked here

Description

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.

History

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.

How it works

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.

In practice

Without quantum fluctuations, galaxies would not have formed, and electronic circuits constantly suffer from thermal noise — voltage fluctuations due to random electron motion.

Fun fact

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.

Concept graph →

Formulas

$$R = \frac{\Phi_1}{\Phi_2}$$
A ratio of two fluxes characterizes a physical condition.
What is in the formula
RvariableFlux ratio — flux ratio
Φ₁variableFirst flux — radiant flux watt
Φ₂variableSecond flux — radiant flux watt
r=\frac{\phi_{{мягк.}}}{\phi_{{жёстк.}}} · 2511.08624
$$\langle \eta_i(t) \eta_j(t') \rangle = 2 \gamma k_B T \delta_{ij} \delta(t-t')$$
Relates noise intensity to temperature and dissipation in equilibrium.
What is in the formula
η_i(t)variablenoise term at time t for component i — noise newton
η_j(t')variablenoise term at time t' for component j — noise newton
Tvariabletemperature — temperature kelvin
γvariabledissipation coefficient — dissipation coefficient kilogram per second
δ_ijvariableKronecker delta — kronecker delta
δ(t-t')variableDirac delta function — dirac delta function per second
k_Bconstantrelates energy to temperature — boltzmann constant = 1.380649e-23 joule per kelvin
⟨...⟩operatoraverage over many realizations — ensemble average
\langle\eta_i(t)\eta_j(t')\rangle=2γ_ik_bt_iδ_{ij}δ(t-t') · 2606.29765
s_j(ω)=|χ(ω)|^2n_{{ir}}(ω) · 2607.04115
$$C(\omega,\omega') = \int dk \, H(k;\omega) H^*(k;\omega') P(k)$$
Correlation of fluctuations relates to the power spectrum.
What is in the formula
C(\omega,\omega')variableCorrelation function of fluctuations at two frequencies — correlation function
H(k;\omega)variableTransfer function or response kernel at wavevector k and frequency omega — transfer function
H^*(k;\omega')variableComplex conjugate of the transfer function at frequency omega' — transfer function
P(k)variablePower spectrum of the underlying fluctuations — power spectrum
\int dkoperatorIntegration over all wavevectors — integral operator
c(w,w')=\langleδf(w)δf^*(w')\rangle=\frac{1}{2π}\int_0^\inftykdkh(k;w)h^*(k;w')p_\kappa(k) · 2607.11887
$$\varepsilon = \frac{|I_{xx} - I_{yy}|}{I_{zz}}$$
Ellipticity measures deviation from axial symmetry.
What is in the formula
εvariableellipticity — ellipticity
I_xxvariablemoment of inertia about x-axis — moment of inertia kilogram metre squared
I_yyvariablemoment of inertia about y-axis — moment of inertia kilogram metre squared
I_zzvariablemoment of inertia about z-axis — moment of inertia kilogram metre squared
\varepsilon=\frac{|i_{xx}-i_{yy}|}{i_{zz}} · 2606.05082v1

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