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The dynamic structure factor is the space-time Fourier transform of an equilibrium two-point correlation function. It resolves fluctuations by angular frequency and wavevector and is related to the dissipative part of a retarded Green function by the fluctuation-dissipation theorem.

For translational drag coefficient $\zeta$ and mobility $1/\zeta$, thermal equilibrium requires $D=k_{B}T/\zeta$. This connects random displacement to dissipative response and is a form of the fluctuation-dissipation theorem. Combining it with Stokes drag law gives the Stokes–Einstein relation.

The fluctuation-dissipation theorem relates equilibrium fluctuations to dissipative linear response. The ordering and normalization of a quantum correlation function matter. For a Hermitian operator $A$ in thermal equilibrium, defineFor nonzero frequency, with spectral function $\rho (\omega )=-2\mathrm{Im}{G}_R(\omega )$ and $\beta =1/(k_{B}T)$,Here $k_B$ is the Boltzmann constant and $\hslash$ is the reduced Planck constant. In units $k_B=\hslash =1$, this reduces to $S^{>}(\omega )=\rho (\omega )/(1-e^{-\omega /T})$.

To see the origin, the Kubo--Martin--Schwinger condition gives $S^{<}(\omega )=e^{-\beta \hslash \omega }{S}^{>}(\omega )$ for the reversed ordering. The commutator spectral function obeys $\hslash \rho =S^{>}-S^{<}$, which proves the displayed relation. The symmetrized spectrum instead satisfiesAt $|\hslash \omega |\ll k_{B}T$, both forms approach $k_{B}T\rho (\omega )/\omega$. This connects a measured noise spectrum with the dissipative part of the corresponding response, after matching the experimental spectrum's ordering and Fourier conventions. ^[ref]

The Kubo formula expresses a linear response through an equilibrium correlation function. For a perturbation $H^{\prime }(t)=-f(t)B$ and observable $A$, one convention gives $\delta ⟨A(t)⟩={\int }_{-\infty }^t{\chi }_{AB}(t-t^{\prime })f(t^{\prime })d{t}^{\prime }$, withThe step function enforces causality, and the commutator records the quantum response.

The sign depends on the declared perturbing-Hamiltonian convention. Applying the framework to currents or polarization relates transport coefficients to fluctuations in an unperturbed system, forming a foundation of linear response theory and the fluctuation-dissipation theorem.^[ref]

Leticia Cugliandolo studies the slow nonequilibrium dynamics of spin glass systems and other complex systems. Her theoretical treatment distinguishes equilibrium relaxation from aging in nonequilibrium systems, where a response depends on how long the system has already been evolving. Two-time correlation functions and responses can violate an equilibrium fluctuation-dissipation theorem, leading to carefully defined effective-temperature descriptions in suitable regimes. Solvable mean-field models connect this dynamics with a complicated free-energy landscape, supplying tests of which features extend beyond a particular material.^[ref]

A physicist researching spin glasses and statistical physics at Sorbonne University.^[ref]

For a dilute spherical no-slip probe of radius $a$ in an isotropic incompressible linear medium at equilibrium, the bead's displacement per unit step force is $J_b(t)=J(t)/(6\pi a)$, where $J$ is the bulk shear creep compliance. The fluctuation-dissipation theorem gives one-coordinate displacement variance $2k_{B}T{J}_b(t)$. Adding the three equal Cartesian contributions proves the displayed relation. In a Newtonian fluid, $J=t/\eta$, so it becomes $⟨\Delta r^2⟩=6Dt$ with the Stokes–Einstein relation $D=k_{B}T/(6\pi \eta a)$.

The particle must sample a continuum response on its scale and remain in thermal equilibrium. A two-dimensional image records only two of the three displacement components; isotropy then gives $⟨\Delta r^2{⟩}_{3d}=(3/2)⟨\Delta r^2{⟩}_{2d}$. Localization noise, confinement and unresolved fast relaxation can otherwise be mistaken for elastic response.

The fluctuation-dissipation theorem for mobility $\tilde{M}$ requiresComparison with the stated noise amplitude givesThis choice makes the stationary Fokker-Planck equation have the Boltzmann distribution proportional to $e^{-V/(k_{B}T)}$.

Equation (1) is Underdamped Langevin dynamics for a unit-mass particle in potential $U$, coupled to a heat bath of temperature $T$. The coefficient $\gamma >0$ is viscous friction, and the noise amplitude is fixed by the fluctuation-dissipation theorem. Its Fokker-Planck equation isFor $\gamma \gg 1$, velocity relaxes rapidly, so formallyThus the overdamped Langevin dynamics isand its density obeys

Thermal molecular motion causes the coarse droplet radius to fluctuate as well as drift down $F(R)$, so its effective equation must be an overdamped Langevin dynamics. With $⟨\Lambda (t)\Lambda (t^{\prime })⟩=\delta (t-t^{\prime })$, detailed balance with equilibrium density proportional to $e^{-F/(k_{B}T)}$ imposes the fluctuation-dissipation theorem. The Model A fluctuation-dissipation relation givesThe factor of mobility ensures that the diffusion in $R$ and the dissipative drift have the same equilibrium Gibbs distribution.

The fluctuation-dissipation theorem in this convention readsIn the classical limit $|\omega |\ll T$, this becomesThe inverse temporal Fourier transform is therefore

For a damped ferromagnetic spin wave, shifting both poles into the lower half-plane preserves causality:Its spectral function isThe classical fluctuation-dissipation theorem then givesHence the zero-frequency static structure factor iswhich has the stated proportionality after absorbing the normalization $4/s$.

For a no-slip spherical particle in a Newtonian fluid at low Reynolds number, Stokes law gives $\zeta =6\pi \eta a$. The fluctuation-dissipation theorem then gives the Stokes–Einstein relation $D=k_{B}T/\zeta$. Using the conventional radial Brownian displacement in three dimensions,Here $k_B$ is the Boltzmann constant, $T$ absolute temperature, $\eta$ dynamic viscosity, $a$ particle radius and $t$ elapsed time. Larger thermal forcing raises diffusion; larger viscous drag reduces it. The formula requires times long relative to momentum relaxation and neglects walls, interactions and external drift.

The word “average” needs a statistic: the vector mean is zero, while the mean radial displacement for Gaussian three-dimensional diffusion is $⟨|\Delta \mathbf{r}|⟩=4\sqrt{Dt/\pi }$, not $\sqrt{6Dt}$. Both scale as $t^{1/2}$. Neither is the total length of a microscopic random path.

Johnson-Nyquist noise is equilibrium electrical fluctuation caused by thermal agitation of charge carriers. A resistor can generate a fluctuating terminal voltage even when its mean current is zero. In the classical frequency range $hf\ll k_{B}T$, the fluctuation-dissipation theorem relates this noise to dissipative resistance. The one-sided voltage power spectral density of an unloaded resistor isEquivalently its short-circuit current spectrum is $4k_{B}T/R$. Over a flat bandwidth $B$ the rms voltage is $\sqrt{4k_{B}TRB}$. A real measurement instead integrates $S_V(f)|H(f){|}^2$ over positive frequency, with loading and the filter's effective noise bandwidth included. A flat spectrum does not imply an infinite physically measurable variance; the circuit and the quantum high-frequency regime limit the ideal white-noise model.

Cooling or narrowing the accepted bandwidth reduces thermal noise. Reducing resistance lowers its open-circuit voltage noise but raises its short-circuit current noise, so the appropriate choice depends on source impedance and amplifier loading. Thermal agitation is not a fluctuating externally applied voltage that can be removed by perfect shielding alone.

Shot noise arises from discrete transfer events. In the simplest shot noise counting model, equal charges $q$ arrive independently at constant rate $r$. A gate of duration $t$ contains a Poisson number $K$, with mean and variance $rt$. Therefore mean current is $I=qr$, and the averaged-current variance is $qI/t$. The corresponding one-sided low-frequency spectrum isfor a flat accepted bandwidth. Electron current has $q=e$. Vacuum emission or photodetection can approximate independent arrivals; correlations in a conductor can suppress or enhance the fluctuations. Discreteness by itself does not guarantee Poisson statistics.

For a fixed mean current the ideal shot-noise formula is not proportional to temperature, whereas Johnson noise is. Increasing current raises absolute shot noise but lowers its fractional size, since $\sqrt{2qIB}/I\propto I^{-1/2}$. Both can appear approximately white within a useful frequency band, and independent noise powers add after referring them to a common measurement point. Their spectra must use the same convention: two-sided densities are half these one-sided values. A real low-current experiment can also contain amplifier noise, low-frequency drift and correlated transfer noise, so a mean current or resistance alone does not specify its complete noise budget.

The fluctuation-dissipation theorem relates spontaneous fluctuations in thermal equilibrium to the dissipative part of the linear response to a weak conjugate force. For an overdamped trapped coordinate, the Langevin equation isThe fluctuation-dissipation relation for a Langevin particle fixes the noise strength from the same drag coefficient $\gamma$ that dissipates motion. With a drive proportional to $e^{-i\omega t}$, the linear response function is $\chi (\omega )=1/(K-i\gamma \omega )$. The two-sided angular-frequency power spectral density then obeysIts integral $\int S_{xx}d\omega /(2\pi )=k_{B}T/K$ agrees with the equipartition theorem. Without a restoring force, the same noise-drag relation gives the diffusion-mobility relation $D=k_{B}T/\gamma$.

In thermal calibration of an optical trap, record a bead's spontaneous displacement fluctuations and compare their spectrum with its response to a weak calibrated oscillatory force. The overdamped harmonic-trap power spectrum and the dissipative response must be consistent with the same temperature and drag. The equilibrium and weak-drive conditions are essential: an actively driven bath need not obey this equilibrium relation.

Passive microrheology uses thermal bead motion as a probe of mechanical compliance. The fluctuation-dissipation theorem relates equilibrium displacement fluctuations to the same linear response measured by a small applied force. The stated thermal stress scale has units of energy per bead volume; dividing the mean-square displacement by $a^2$ provides a dimensionless effective strain. With that prescribed analogy,The displacement is relative to the starting position, and the variance here is the sum over three spatial coordinates. For a Newtonian fluid, substitution gives $⟨\Delta r^2⟩=k_{B}Tt/(\pi \eta a)=6Dt$, exactly the Stokes–Einstein relation $D=k_{B}T/(6\pi \eta a)$.

The continuum interpretation assumes a dilute no-slip spherical probe, thermal equilibrium, linear response and a homogeneous incompressible medium on the bead scale. The stated stress estimate is an effective scale for the analogy; a literal instantaneous stress variance would also require a microscopic or temporal averaging prescription.

A moving or externally forced colloid disturbs the surrounding fluid, changing the velocity of other colloids even without direct contact. These hydrodynamic interactions are long-ranged at small Reynolds number. In an unbounded incompressible Newtonian liquid, the Stokes flow response to a point force is the Stokeslet:Its Oseen tensor isThis is a mobility tensor, with units velocity per force, rather than a stress tensor. At separation $r$ the response parallel to the line of centres is $1/(4\pi \eta r)$; the perpendicular response is $1/(8\pi \eta r)$. Flow is not isotropic even though the solvent is. The disturbance decays as $1/r$, so treating all drag forces as independent can miss correlated motion.

For well-separated equal spheres of radius $a$, the leading hydrodynamic mobility matrix givesThe first term is the isolated-sphere self mobility; the remaining terms couple distinct particles. At thermal equilibrium the fluctuation-dissipation theorem also predicts correlated short-time increments, $⟨\Delta r_{i\alpha }\Delta r_{j\beta }⟩\simeq 2k_{B}T{\mu }_{i\alpha ,j\beta }\Delta t$ with the configuration locally fixed. Thus correlated Brownian displacements can reveal hydrodynamic coupling without an applied force.

The point-force approximation requires $r\gg a$ and negligible inertia. It diverges at the origin and cannot replace the finite self mobility. Near-contact lubrication, walls, many-body corrections, and a viscoelastic solvent require more complete mobility laws. A force-free particle can instead have a faster-decaying leading disturbance, so the $1/r$ claim specifically concerns a net-force-driven object.

Equipartition gives mean energy $k_{B}T/2$ for each independent quadratic term in a classical canonical Hamiltonian. For example, $p^2/(2m)$ and $k{x}^2/2$ each contribute this amount in a harmonic oscillator. It is a thermal average, not an instantaneous equality; constraints, nonquadratic terms or quantum level spacings large compared with $k_{B}T$ require a different calculation.

The fluctuation-dissipation theorem connects equilibrium fluctuations with the dissipative part of linear response. If an applied force couples as $-f(t)x$, use response to $e^{-i\omega t}$ and define the two-sided angular-frequency spectrum $S_{xx}(\omega )=\int e^{i\omega t}⟨\delta x(t)\delta x(0)⟩dt$. The classical relation isFor a viscously damped particle, $m\dot{v}=-\zeta v+\xi (t)$, the corresponding white-noise force satisfies $⟨\xi (t)\xi (t^{\prime })⟩=2\zeta k_{B}T\delta (t-t^{\prime })$. Noise replenishes energy lost by friction; their strengths cannot be varied independently at fixed equilibrium temperature.

Electrically, the one-sided voltage-noise spectrum per hertz of an equilibrium resistor is $4k_{B}TR$. A capacitor in parallel filters it toThe same equilibrium voltage variance follows from equipartition. Increasing R makes the spectrum taller and narrower without changing its integrated area. This thermodynamic connection between dissipative conductance and thermal electrical fluctuations is the content of the classical noise calculation.^[ref]

Brownian motion is the irregular motion of a mesoscopic particle caused by thermal molecular impacts. A colloidal sphere in a fluid provides a physical example. The random force competes with dissipative drag; averaging many trajectories produces diffusion even though each trajectory fluctuates.

A free-particle Langevin equation with thermal Langevin force normalization isThe noise has zero mean, and its strength is linked to drag by the fluctuation-dissipation theorem. The velocity relaxes on the time scale $\tau =m/\zeta$. At long times $t\gg \tau$, the Einstein diffusion relation isFor a slowly moving no-slip sphere of radius $a$ in a continuum fluid of dynamic viscosity $\eta$, Stokes drag law gives $\zeta =6\pi \eta a$. The particle probability density obeys ${\partial }_{t}P=D{\nabla }^{2}P$; a localized initial position develops a Gaussian profile. The ballistic-to-diffusive Brownian displacement changes character: at $t\ll \tau$, equilibrium velocities instead give ballistic mean-square displacement $3(k_{B}T/m)t^2$.

The drag-noise model assumes a classical bath with a sufficiently short correlation time; a mathematical white-noise trajectory does not resolve individual molecular collisions. Tracking nonliving colloidal particles measures diffusion and tests the connection between thermal fluctuations and dissipation.^[ref]