Hydrodynamic Fluctuations and Noise

Thermal hydrodynamic fluctuations are stochastic fluctuations of the same conserved densities and currents that appear in deterministic constitutive relations. In equilibrium, fluctuation–dissipation fixes their two-point kernel from the dissipative transport matrix. White Gaussian noise is only the leading Markovian, classical, coarse-grained limit; nonlinear variables, causal completions, quantum frequencies, and finite regulators generally produce multiplicative, colored, non-Gaussian, or bilocal structures.

Required background. KMS Relations and Fluctuation–Dissipation fixes the equilibrium noise–response relation. Relativistic Dissipative Hydrodynamics supplies the transport coefficients.

Helpful background. Strong Hyperbolicity, Stability, and Causal Propagation and BDNK First-Order Causal Hydrodynamics separate causal PDE structure from infrared transport. Multiplicative, Colored, and Conserved Noise develops stochastic prescriptions.

Conserved diffusion with matched noise

Let $n$ be one charge density, $\chi=\partial n/\partial\mu$ its static susceptibility, and $\sigma$ its conductivity. At fixed temperature,

$$\mathbf j=-\sigma\boldsymbol\nabla\mu+\boldsymbol\xi =-D\boldsymbol\nabla n+\boldsymbol\xi, \qquad D=\frac{\sigma}{\chi}.$$

The continuity equation becomes

$$\partial_t n-D\nabla^2n=-\boldsymbol\nabla\!\cdot\boldsymbol\xi.$$

In the local classical equilibrium and Markov limits, fluctuation–dissipation requires

$$\left\langle\xi_i(t,\mathbf x)\xi_j(t',\mathbf x')\right\rangle = 2T\sigma\,\delta_{ij}\, \delta(t-t')\delta^{(d_s)}(\mathbf x-\mathbf x').$$

Solving in Fourier space gives the symmetric density correlator

$$G^S_{nn}(\omega,k) = \frac{2T\sigma k^2}{\omega^2+(Dk^2)^2} = \frac{2T\chi Dk^2}{\omega^2+(Dk^2)^2}.$$

Its equal-time integral is the independent check:

$$\int_{-\infty}^{\infty}\frac{\mathrm d\omega}{2\pi} G^S_{nn}(\omega,k)=T\chi.$$

The same diffusion normalization and its relation to retarded response are derived in Kovtun 2012, §§2.3 and 3.1, Open PDF.

The noise is a current, rather than an additive density source, so exact conservation survives realization by realization. Landau and Lifshitz give the corresponding stochastic stress tensor; in a local rest frame its shear and bulk covariance is fixed by $2T\eta$ and $2T\zeta$ with the same tensor projectors as the viscous stress Landau and Lifshitz 1987, §XVII, pp. 537–545.

When white Gaussian noise is justified

The delta functions approximate a microscopic kernel whose correlation time and length are unresolved. They require

$$\omega\tau_{\mathrm{noise}}\ll1, \qquad k\ell_{\mathrm{noise}}\ll1,$$

and a cell containing enough microscopic degrees of freedom for Gaussian closure. The continuum variance at one point diverges: $\delta^{(d_s)}(0)$ is not an observable. A lattice spacing, smoothing kernel, or momentum cutoff must be declared, and bare thermodynamic and transport parameters then depend on it.

If $\sigma$, $T$, or the hydrodynamic measure depends on the fluctuating fields, the noise is multiplicative. Itô, Stratonovich, kinetic, or a time-symmetric prescription changes drift and Jacobian terms. The prescription is part of the theory, not a numerical option. Non-Gaussian cumulants arise beyond the quadratic effective action and are constrained, but not generally fixed, by equilibrium symmetry.

At quantum frequencies the classical factor $2T$ is replaced by the KMS frequency kernel. The symmetrized noise is colored even in equilibrium, schematically through $\omega\coth(\beta\omega/2)$; the white expression is its $\lvert\omega\rvert\ll T$ limit.

Causal formulations change the kernel

Consider a relaxing current,

$$\tau\partial_t\mathbf j+\mathbf j =-D\boldsymbol\nabla n+\boldsymbol\xi.$$

White forcing of this enlarged Markov system produces an effective current spectrum proportional to

$$\frac{1}{1+\omega^2\tau^2}$$

after the transient current is eliminated. Thus a local differential representation can become colored and temporally nonlocal in a density-only representation. Noise variables, response kernel, and initial covariances must be transformed together.

This point is sharper for BDNK. A linear stochastic BDNK construction generally has a bilocal Martin–Siggia–Rose action and nonwhite primary-field noise; importing the local Landau–Lifshitz tensor unchanged is not consistent. Conserved-density correlators can nevertheless agree with a related transient formulation Gavassino, Mullins, and Hippert 2024, §§III–VI, Open PDF. The result is a linear equilibrium construction, not a universal nonlinear stochastic BDNK theorem. The formulation-sensitive statement here was checked through 10 August 2026; later nonlinear results must be compared using the same variables, regulator, and noise kernel.

Fluctuating hydrodynamics and SK consistency reference

This table is the canonical check for the fluctuating and Schwinger–Keldysh pages.

Layer	Required declaration	Decisive check	Failure if omitted	Strongest licensed conclusion
Noise covariance	Variables, frame, $T$, susceptibilities, transport matrix	Equal-time covariance and fluctuation–dissipation	Wrong equilibrium measure	Gaussian two-point fluctuations in the stated regime
Regulator and measure	Cell size or cutoff, field measure, Jacobian	Cutoff variation with parameter rematching	Pointwise divergences or measure-dependent drift	Regulated EFT prediction, not cutoff-free microscopic noise
Stochastic prescription	Itô, Stratonovich, kinetic, or contour definition	Stationary distribution and induced drift	Multiplicative-noise ambiguity	Well-defined stochastic evolution
SK normalization and reality	$Z[J,J]=1$, conjugation rule	Action vanishes at $a=0$ and obeys SK reality	Violation of unitarity normalization	Consistent doubled generating functional
Causal $r/a$ structure	Retarded inverse operator and initial conditions	No response before the source; all-$a$ causal zeros	Advanced contamination	Retarded response within the EFT
Thermal symmetry	State, antiunitary map, thermal vector and twist	KMS Ward identity and low-frequency FDT	Unjustified equilibrium relations	Local-equilibrium constraints only
Positivity	Imaginary quadratic kernel	Nonnegative noise covariance	Nonconvergent path integral or negative variance	Semipositive Gaussian sector
Power counting and matching	Gradient, $a$-field, loop, amplitude, and frame order	Reproduce constitutive poles and Kubo coefficients	Double counting or unmatched vertices	Equivalence through the retained order
Renormalization	Bare versus renormalized EOS and transport	Cutoff-independent long-distance observable	Regulator-dependent “prediction”	Scale-qualified transport data
Causal completion and cutoff	Transient/BDNK formulation and UV stop rule	Characteristics plus transformed noise kernel	White-noise excitation beyond validity	Causal stochastic claim only under its formulation-specific hypotheses

The table distinguishes requirements that follow from SK normalization from the additional thermal assumptions behind KMS. The next two pages derive that separation.

The upper-left branch places stochastic hydrodynamics relative to the other extensions. Inspect its two-part label: SK structure organizes response and noise, while dynamical KMS requires the additional equilibrium hypothesis encoded in this page’s consistency table.

Stochastic currents supplement the ordinary conserved-density theory with fluctuations whose covariance is fixed by dissipative response in equilibrium. An SK formulation supplies normalization, reality, and positivity constraints; dynamical KMS is the extra thermal condition behind fluctuation–dissipation. The other branches require different added variables and must not be generated by reinterpreting this noise sector. The diagram is schematic.

In text: choose a coarse-graining regulator, add conserved noise with covariance $2T\sigma$, verify the equal-time susceptibility $T\chi$, and distinguish white, colored, and multiplicative noise. This branch extends ordinary hydrodynamics without adding a new conserved density, whereas the other branches shown do.

Exercise

Verify the equal-time variance above and show what fails if the noise strength is written $2T\chi$ instead of $2T\sigma$.

Solution

Using

$$\int_{-\infty}^{\infty}\frac{\mathrm d\omega}{2\pi} \frac{1}{\omega^2+a^2}=\frac{1}{2a},$$

with $a=Dk^2$ gives $T\chi$. Replacing $\sigma$ by $\chi$ gives $T\chi/D$, which has the wrong value and, unless units are hidden in the definition of the noise, the wrong dimensions. The noise pairs with the dissipative mobility $\sigma$; susceptibility converts that mobility into $D$.

Common pitfalls

Removing conservation with additive density noise. Conserved noise enters through a divergence unless the source itself represents explicit charge exchange.

Taking the continuum delta function literally. Local variances require a regulator and renormalized parameters.

Attaching Navier–Stokes noise to a causal theory. The response variables and fluctuation kernel must be derived in the same formulation and frame.

Where this leads

Schwinger–Keldysh Effective Actions for Fluids generates the diffusion equation and this noise covariance from one action. Long-Time Tails and Fluctuation Renormalization then shows why the nonlinear theory is not exhausted by Gaussian noise.

References

• Gavassino, Lorenzo, Nicki Mullins, and Mauricio Hippert. 2024. “Consistent Inclusion of Fluctuations in First-Order Causal and Stable Relativistic Hydrodynamics.” Physical Review D 109: 125002. DOI. Open PDF.

• Kovtun, Pavel. 2012. “Lectures on Hydrodynamic Fluctuations in Relativistic Theories.” Journal of Physics A: Mathematical and Theoretical 45: 473001. DOI. Open PDF.

• Landau, L. D., and E. M. Lifshitz. 1987. Fluid Mechanics. 2nd ed. Course of Theoretical Physics, vol. 6. Pergamon Press. DOI.