Hydrodynamics and Effective Theory Preview

Hydrodynamics is the long-wavelength, late-time effective theory of conserved densities and any other fields whose relaxation is parametrically slow. Its domain is much wider than the mechanics of ordinary liquids. Energy diffusion in a spin chain, charge flow in an electron fluid, sound in a neutral gas, viscous momentum transport, superfluid phase motion, and critical relaxation can all have hydrodynamic descriptions when the correct slow variables are retained.

The organizing equation is exact:

$$\partial_t n_A + \partial_i j_A^i = 0.$$

It states that the local density $n_A$ of a conserved charge changes only by current flow. It does not determine the current. The effective-theory step is a constitutive expansion,

$$j_A^i = j_A^i \left[ n_B,T,u^j,\ldots \right],$$

ordered in amplitudes, derivatives, and fluctuations. Combining exact conservation with approximate constitutive relations produces diffusion, shear relaxation, sound, and coupled transport.

Hydrodynamics is predictive when local equilibration and a regular long-wavelength current expansion make the retained fields slow. Its coefficients and field content depend on the state, symmetry, phase, and scale window.

Helpful background. Density and current operators supplies continuity equations; transport coefficients derives the response functions used to determine diffusion and viscosity.

Hydrodynamic Densities and Response Conventions

Unless stated otherwise, use natural units:

$$\hbar = c = k_{\mathrm B} = 1.$$

Spatial dimension is $d$. Repeated spatial indices are summed. The Fourier convention is

$$X(\omega,\mathbf q) = \int dt \int d^d\mathbf x\, e^{i\omega t-i\mathbf q\cdot\mathbf x} X(t,\mathbf x).$$

Therefore

$$\partial_t \longrightarrow -i\omega, \qquad \partial_i \longrightarrow iq_i.$$

Retarded poles of stable modes lie in the lower half of the complex $\omega$ plane. A diffusive pole is

$$\omega = -iDq^2 +\cdots, \qquad D\gt0.$$

The static susceptibility matrix is

$$\chi_{AB} = \left( \frac{\partial n_A} {\partial\mu_B} \right)_{\mathrm{eq}},$$

with the other thermodynamic variables held fixed according to the stated ensemble. A different ensemble or choice of independent densities changes the matrix that must be inverted.

Why Conservation Creates Slow Motion

Let

$$Q_A = \int_V d^d\mathbf x\, n_A(\mathbf x)$$

be a conserved charge in a periodic or closed system. A long-wavelength modulation cannot disappear by a local decay of $Q_A$. It must move through space. For local, regular constitutive relations, spatial derivatives supply small factors of $q$, making the corresponding frequencies or decay rates vanish as $q\to0$. Locality and regularity matter: an unscreened Coulomb field can produce a singular current response and a gapped plasma oscillation even though total charge is conserved. Global conservation alone does not make every density-response pole gapless.

This is the robust reason hydrodynamic poles approach the origin. It does not require weak coupling or long-lived microscopic quasiparticles. Strongly interacting systems can be hydrodynamic when local nonconserved structure relaxes rapidly compared with the conserved fields.

The converse is not automatic. A conserved quantity need not produce ordinary diffusion:

• momentum conservation can combine with density to produce sound;
• ballistic conserved overlaps can generate a Drude contribution;
• long-range interactions can change the spatial scaling;
• integrability can supply an extensive family of slow charges;
• localization can obstruct transport;
• a broken continuous symmetry adds a Goldstone phase;
• a critical order parameter can become slow even when it is not conserved.

The slow fields must therefore be identified before constitutive equations are truncated.

Selecting Hydrodynamic Fields

The minimal candidates are local densities of exact conserved quantities:

$$\epsilon, \qquad n_A, \qquad \pi_i.$$

Here $\epsilon$ is energy density, $n_A$ denotes internal charges, and $\pi_i$ is momentum density when momentum is locally conserved. Crystal momentum in a lattice is not automatically an exactly conserved hydrodynamic density: Umklapp, impurities, substrates, and boundaries can relax it.

Additional slow fields are required when they have a parametrically long lifetime:

Physical structure	Additional field
broken continuous symmetry	Goldstone phase
crystal or viscoelastic solid	displacement or strain
near a continuous transition	critical order parameter
weakly broken symmetry	approximately conserved density
integrable one-dimensional system	quasiparticle density over rapidity
chemical reaction or imbalance	slow reaction coordinate

A field should not be retained merely because it is easy to name. It should be supported by a conservation law, symmetry, small relaxation rate, or demonstrated spectral pole. Conversely, integrating out a slow field usually makes the remaining equations nonlocal in time.

Projection viewpoint

One can regard hydrodynamics as a projection onto a slow subspace. Fast microscopic operators are not set to zero. Their effects survive through:

• an equation of state;
• static susceptibilities;
• transport and relaxation coefficients;
• noise kernels;
• higher-derivative operators;
• renormalization of the retained parameters.

This is why two microscopically different systems can share hydrodynamic equations while having different numerical coefficients.

Local Equilibrium

A leading local-equilibrium ansatz has the schematic form

$$\begin{aligned} \rho_{\mathrm{LE}} &= \frac{1}{Z_{\mathrm{LE}}} \exp \left[ -\int d^d\mathbf x\, \beta(\mathbf x) \mathcal K(\mathbf x) \right], \\ \mathcal K &= \epsilon -\mu_A n_A -u^i\pi_i. \end{aligned}$$

The fields $\beta(\mathbf x)$, $\mu_A(\mathbf x)$, and $u^i(\mathbf x)$ are local thermodynamic multipliers. This expression is an organizing approximation, not a claim that an inhomogeneous evolving state is exactly Gibbsian at every point. Operator ordering, boundary terms, gauge fields, and relativistic covariance require more careful formulations in a full treatment.

Local equilibrium means that nonhydrodynamic information relaxes over scales

$$\ell_{\mathrm{micro}}, \qquad \tau_{\mathrm{micro}},$$

short compared with the observation scales. Then a small cell can be assigned thermodynamic variables while those variables vary slowly from cell to cell.

The required hierarchy is

$$q\ell_{\mathrm{micro}} \ll 1, \qquad \omega\tau_{\mathrm{micro}} \ll 1.$$

It does not assert that the whole system is in global equilibrium. A temperature gradient, velocity profile, or slowly varying chemical potential is precisely what hydrodynamics evolves.

Microscopic Constraints and Material Coefficients

Conservation laws, symmetries, and Ward identities survive coarse graining. Equations of state, transport coefficients, relaxation scales, and noise amplitudes must be calculated or measured for the material and state. Universality concerns the allowed long-distance structure; it does not imply a universal numerical viscosity, conductivity, or diffusion constant.

Exact Equations and Constitutive Input

For several conserved scalar densities,

$$\partial_t n_A + \partial_i j_A^i = 0.$$

This equation can hold as an operator identity. Hydrodynamics closes it by expressing $j_A^i$ in terms of the retained fields. Near a homogeneous state with no background flow,

$$j_A^i = -L_{AB} \partial_i\mu_B +\cdots,$$

Here the temperature is fixed and $\mu_B$ is the local chemical-potential departure from equilibrium. This defines the normalization of $L$: if one instead writes the force as $\nabla(\mu_B/T)$, its coefficient is $T L$ at uniform $T$. Mixing these two conventions would give a spurious factor of temperature in the diffusion matrix. The ellipsis contains higher gradients, nonlinearities, reversible terms, background-field terms, and noise.

The constitutive relation is not exact because the current is itself a microscopic operator with fast components. It becomes useful after those components have relaxed and their residual effect can be represented by local coefficients.

Conservation with sources and sinks

Background gauge fields or explicit symmetry breaking modify the balance equation. Schematically,

$$\partial_t n_A + \partial_i j_A^i = s_A - \Gamma_{AB}\delta n_B.$$

The source $s_A$ describes injection or extraction. A small relaxation matrix $\Gamma_{AB}$ describes weakly broken conservation. Neither term should be inserted without identifying the microscopic process and its scaling.

Derivative and Amplitude Expansions

Hydrodynamics is commonly organized in derivatives:

$$j_A^i = j_{A,(0)}^i + j_{A,(1)}^i + j_{A,(2)}^i +\cdots.$$

The ideal term $j_{(0)}$ contains no gradients. First-order terms include diffusion, viscosity, and heat conduction. Higher orders encode finite relaxation times, dispersive corrections, and additional tensor structures.

Linearized hydrodynamics also expands in small amplitudes:

$$n_A = n_{A,0} + \delta n_A, \qquad |\delta n_A| \ll n_{A,\mathrm{scale}}.$$

The positive scale $n_{A,\mathrm{scale}}$ characterizes how far the equation of state can be linearized; it need not equal $|n_{A,0}|$, which can vanish for a neutral background. These are independent approximations. A slowly varying flow can be nonlinear in amplitude; a small-amplitude perturbation can vary too rapidly for hydrodynamics. Turbulent or shock-like regimes require nonlinear equations and may invalidate a simple derivative truncation.

Mode-dependent power counting

For sound,

$$\omega \sim q,$$

while for diffusion,

$$\omega \sim q^2.$$

Terms of the same apparent derivative order can therefore contribute differently in different channels. A consistent calculation declares its scaling before deciding which temporal and spatial derivatives to retain.

Diffusion as the Minimal Example

For one conserved scalar density in an isotropic state,

$$\mathbf j = -D\nabla n$$

gives

$$\partial_t n = D\nabla^2 n.$$

A plane wave then has

$$\omega_{\mathrm{diff}} = -iDq^2.$$

The vanishing decay rate at $q=0$ reflects exact conservation. The detailed density response,

$$\chi_{nn}^R(\mathbf q,\omega) = \chi \frac{Dq^2} {Dq^2-i\omega},$$

uses the physical susceptibility $\chi^R_{AB}(t)=+i\theta(t)\langle[A(t),B]\rangle$ for a perturbation $H\prime=-fB$. Its noncommuting static and uniform limits are derived on Transport Coefficients Preview.

Diffusion is not synonymous with incoherence. It is a statement about the long-distance pole. Microscopically, the same $D$ can arise from quasiparticle scattering, chaotic operator dynamics, environmental collisions, or strongly interacting relaxation without sharp carriers.

Momentum, Shear, and Sound

When momentum is conserved, its current is the stress tensor:

$$\partial_t\pi_i + \partial_j\Pi_{ij} = 0.$$

Transverse momentum does not couple to pressure at linear order. In a simple isotropic Galilean-invariant fluid,

$$\omega_{\mathrm{shear}} = -i\nu q^2 +\cdots, \qquad \nu = \frac{\eta}{\rho_m}.$$

Here $\eta$ is shear viscosity and $\rho_m$ is mass density.

Longitudinal density and momentum couple. The ideal linear equations can be written

$$\begin{aligned} \partial_t\delta\rho &= -\rho_0 \nabla\boldsymbol\cdot\mathbf v, \\ \rho_0\partial_t\mathbf v &= -\nabla\delta p, \\ \delta p &= c_s^2\delta\rho. \end{aligned}$$

Here $\rho$ is mass density and $c_s^2=(\partial p/\partial\rho)_{s/n}$ for an adiabatic sound wave. The entropy-per-particle perturbation is held fixed in these three equations; a general thermal perturbation also requires an energy or entropy equation and a heat mode. Dissipative gradients shift the sound poles to

$$\omega_\pm(q) = \pm c_s q - i\Gamma_s q^2 + O(q^3).$$

The coefficient $\Gamma_s$ depends on viscosity, thermal transport, thermodynamic derivatives, dimension, and field content. Its detailed form is not universal and remains with the transport treatment.

If continuous momentum is relaxed by a lattice, substrate, disorder, or bath, ordinary fluid sound and shear need not remain hydrodynamic. Solids can still support phonons because broken translations supply displacement fields. The mechanism selecting the slow field matters more than the name “sound.”

Coupled Diffusion

Suppose several scalar charges are conserved. Linearized constitutive relations lead to

$$\partial_t\delta n_A = D_{AB} \nabla^2\delta n_B.$$

For the isothermal chemical-potential convention above,

$$D = L\chi^{-1},$$

where $L$ is the kinetic matrix defined by $\mathbf j=-L\nabla\boldsymbol\mu$ and $\chi=(\partial\mathbf n/\partial\boldsymbol\mu)_T$. If energy and heat currents are included, use the full thermodynamic Hessian and its conjugate forces rather than importing this isothermal submatrix. The eigenvalues of $D$ determine the diffusive eigenmodes. The original particle, spin, or heat densities need not diffuse independently.

Strict decay at nonzero $q$ requires positive real parts of the diffusion eigenvalues. A zero-real-part eigenvalue is only marginally stable and must not have a nontrivial Jordan block that generates polynomial growth. In time-reversal-invariant equilibrium, Onsager relations constrain $L$ after the parities of the variables are included. Positivity applies naturally to the dissipative symmetric part in the thermodynamic metric, not to every matrix entry separately.

For time-reversal-even charges with symmetric positive $L$ and positive definite $\chi$, the matrix $D$ need not look symmetric. It is similar to $\chi^{-1/2}L\chi^{-1/2}$, so its eigenvalues are real and positive. For example, in compatible units,

$$\chi=\begin{pmatrix}2&0\\0&1\end{pmatrix},\qquad L=\begin{pmatrix}2&1\\1&2\end{pmatrix},\qquad D=\begin{pmatrix}1&1\\1/2&2\end{pmatrix}.$$

The decay constants are $(3\pm\sqrt3)/2$, neither the eigenvalues $1,3$ of $L$ nor the diagonal entries of $D$. Susceptibilities convert chemical-potential gradients into density gradients; omitting that conversion changes the physical modes.

Matching Transport Coefficients

Hydrodynamics predicts pole forms but does not calculate every coefficient from symmetry. Matching supplies:

$$\begin{gathered} p(T,\mu_A), \qquad \chi_{AB}, \qquad L_{AB}, \\ \qquad \eta, \qquad \zeta, \qquad \ldots \end{gathered}$$

Possible matching routes include:

• equilibrium thermodynamics for the equation of state and susceptibilities;
• Kubo or Green–Kubo correlation formulas;
• kinetic theory in a controlled quasiparticle regime;
• exact or numerical real-time evolution;
• experiment;
• matching from a more microscopic effective theory.

The matching scale must lie inside an overlap window: below microscopic excitations that have been integrated out, but above the frequencies and wave numbers where the final hydrodynamic prediction is used.

Order of limits

Transport coefficients often involve

$$\begin{aligned} L&\to\infty, & q&\to0, \\ \omega&\to0, & \eta_{\mathrm{reg}}&\to0. \end{aligned}$$

These limits need not commute. A finite isolated system has a discrete spectrum and recurrences; a hydrodynamic pole requires a bulk and long-time regime. A smooth curve produced by arbitrary numerical broadening is not by itself a transport coefficient.

Kubo Formula derives source response and contact terms. Transport Coefficients Preview explains the coefficients and finite-size limits.

Hydrodynamics as an Effective Field Theory

Hydrodynamics has the usual EFT ingredients:

• a specified set of low-energy fields;
• symmetries and exact constraints;
• a derivative and fluctuation expansion;
• coefficients matched from shorter scales;
• a cutoff beyond which the description is not trusted;
• field redefinitions that change conventions but not observables.

Its unusual feature is dissipation. A single ordinary real-time action does not naturally encode both causal response and fluctuations while preserving normalization. The in-in construction supplies the needed doubled fields.

Introduce background sources on two branches:

$$\begin{aligned} Z[A_+,A_-;g_+,g_-] &= \operatorname{Tr} \left[ U_+ \rho_0 U_-^{\dagger} \right], \\ U_\pm &\equiv U_{A_\pm,g_\pm}. \end{aligned}$$

The gauge sources $A_\pm$ generate conserved currents. Metric or geometric sources $g_\pm$ generate stress tensors. Equal sources give

$$Z[A,A;g,g] = 1.$$

Gauge invariance yields current Ward identities; diffeomorphism or translation structure yields energy-momentum identities. The effective action must realize these constraints after fast fields are integrated out.

A Diffusive Schwinger–Keldysh Action

In the classical thermal regime, a schematic quadratic action for one density is

$$S_{\mathrm{diff}} = \int dt\,d^d\mathbf x \left[ n_a \left( \partial_t n_r - D\nabla^2n_r \right) + iT\chi D \partial_i n_a \partial_i n_a \right].$$

The $r$ field is the physical density fluctuation. Here $n_a$ is an auxiliary response field with the normalization displayed in the action. It is not literally the difference of two density fields: identifying it with a microscopic branch difference requires an additional change of variables and units. Source terms, initial-state terms, and nonlinear completion are omitted here, so this is a structural preview rather than a standalone microscopic action.

The action displays the characteristic constraints:

$$S_{\mathrm{diff}}[n_r,0] = 0,$$

and

$$\operatorname{Im} S_{\mathrm{diff}} = T\chi D \int dt\,d^d\mathbf x\, (\nabla n_a)^2 \geq 0.$$

The first relation expresses the normalized bulk response structure; the initial distribution still has to be supplied. The second damps response-field configurations when $T\chi D\geq0$.

Varying the real part with respect to $n_a$ gives the diffusion equation. The imaginary term is equivalent to a noisy current:

$$\partial_t n_r - D\nabla^2n_r = -\partial_i\xi_i,$$

with

$$\begin{aligned} \langle\xi_i(x)\rangle &= 0, \\ \langle\xi_i(x)\xi_j(x')\rangle &= 2T\chi D\, \delta_{ij} \delta^{(d+1)}(x-x'). \end{aligned}$$

The noise strength is not optional decoration. In equilibrium it is tied to dissipation by fluctuation–dissipation. Removing it while retaining $D$ produces the mean diffusion equation but not the correct equilibrium fluctuations or loop corrections.

For a box-normalized nonzero mode $n_{\mathbf q}=V^{-1/2}\int_V d^dx\,e^{-i\mathbf q\cdot\mathbf x}n_r(\mathbf x)$, write $C_q(t)=\langle|n_{\mathbf q}(t)|^2\rangle$. If the mode initially has no variance, the noisy diffusion equation gives

$$C_q(t)=T\chi\left(1-e^{-2Dq^2t}\right).$$

The equilibrium variance is independent of $q$ at this leading local order, but its approach is slower for longer wavelengths. At exactly $q=0$, both damping and conserved-current noise vanish: the variance of the total charge is fixed by its initial ensemble, not selected by diffusion.

Noisy diffusion with $D=T\chi=1$ and $C_q(0)=0$. Shorter wavelengths equilibrate faster. The gray $q=0$ line stays at zero; conservation preserves its initial distribution. The white-noise description applies within the classical hydrodynamic frequency range.

Different field normalizations move factors of $\chi$, $T$, and two between the action and sources. The physical content is the retarded diffusion operator and the equilibrium noise kernel in compatible conventions.

Unitarity, KMS, and the Second Law

A hydrodynamic in-in effective action for fields $\Psi_r,\Psi_a$ inherits

$$S_{\mathrm{eff}}[\Psi_r,0] = 0,$$

and a reality relation of the form

$$S_{\mathrm{eff}}[\Psi_r,\Psi_a]^* = -S_{\mathrm{eff}}[\Psi_r,-\Psi_a].$$

The imaginary part must have the sign needed for convergence and positivity. In thermal equilibrium, a dynamical implementation of KMS further relates response and noise. These conditions constrain which dissipative terms can appear and connect them to entropy production and Onsager reciprocity.

At the constitutive level, entropy balance has the schematic form

$$\partial_t s + \partial_i j_s^i = X_M L_{MN} X_N \geq 0,$$

where $X_M$ are thermodynamic forces normalized for this entropy-production formula (which can differ from the chemical-potential gradients above). This is the deterministic constitutive statement; individual noisy histories need not have nonnegative entropy production at every instant. Positivity constrains the dissipative symmetric part of $L$. Nondissipative Hall-like or antisymmetric transport can contribute to currents without producing entropy.

The second law is not an independent microscopic proof that every phenomenological coefficient is positive. Its implementation depends on field definitions, equations of motion, anomalies, and allowed improvement terms. Full EFT formulations make these qualifications systematic.

Fluctuating and Nonlinear Hydrodynamics

Hydrodynamic fields fluctuate because a coarse cell contains finitely many microscopic degrees of freedom and exchanges conserved quantities with neighboring cells. Nonlinear constitutive terms couple the slow modes to one another. Schematically,

$$\partial_t\Psi = \mathcal L\Psi + \mathcal N[\Psi,\Psi] + \xi.$$

Loops of hydrodynamic modes can generate:

• algebraic long-time tails;
• nonanalytic dependence on $\omega$ and $q$;
• scale-dependent effective transport coefficients;
• enhanced fluctuations in low spatial dimension;
• corrections that no finite polynomial derivative expansion reproduces term by term.

For a common diffusive mode-coupling channel, a correlation can decay as

$$C(t) \sim t^{-d/2},$$

although the exponent and amplitude depend on the observable and coupled fields. A power-law tail is therefore a prediction of the nonlinear low-energy theory, not automatically a signature of a microscopic quasiparticle.

Hydrodynamic fluctuations are part of the EFT’s dynamics. Treating constitutive equations as noiseless deterministic laws is sufficient for mean linear response in some regimes, but not for fluctuation spectra, higher correlations, or loop-level precision.

Hydrodynamic Frames and Field Redefinitions

Away from strict equilibrium, temperature, chemical potential, and velocity are definitions of effective fields. At derivative order they can be redefined:

$$\begin{aligned} T &\to T+\delta T_{(1)}, \\ \mu_A &\to \mu_A+\delta\mu_{A,(1)}, \\ u^i &\to u^i+\delta u_{(1)}^i. \end{aligned}$$

Such transformations move terms among the constitutive relations. Choosing a Landau, Eckart, mass, entropy, or another frame is a convention for the fields, not a change in the underlying physical state.

Observable currents, response functions, pole locations, and entropy production must be frame invariant to the working order. Comparing two papers coefficient by coefficient before translating frames can create false disagreements.

The EFT analogy is direct: operators proportional to lower-order equations of motion can often be shifted by field redefinitions. A nonredundant basis is defined only after those equivalences are handled.

Approximate Conservation and Quasihydrodynamics

If a charge fluctuation $n$ about its equilibrium value relaxes slowly,

$$\partial_t n + \nabla\boldsymbol\cdot\mathbf j = -\Gamma n, \qquad \Gamma\tau_{\mathrm{micro}} \ll 1,$$

then its mode becomes

$$\omega = -i\Gamma - iDq^2 +\cdots.$$

It is not exactly hydrodynamic because the pole does not reach the origin at $q=0$, but it remains parametrically slow. Keeping the field explicitly can be much more local than integrating it out.

For a classical equilibrium scalar mode, the extra relaxation also needs noise of strength $2T\chi\Gamma$ to preserve its equilibrium variance; conserved-current noise alone is insufficient. Examples include weak momentum relaxation, intervalley imbalance, nearly conserved spin, weakly broken integrability, and slow chemical reactions. The relaxation matrix and constitutive coefficients must be expanded in a consistent hierarchy; taking $\Gamma$ large while retaining the field defeats the quasihydrodynamic premise.

Extra Fields in Ordered and Critical Phases

Broken continuous symmetry

A superfluid or magnet with a broken continuous symmetry carries a Goldstone field in addition to conserved densities. The phase can support propagating modes and nondissipative currents. Goldstone Modes in Many-Body Systems explains the symmetry and counting theory.

Superfluidity in Condensed Matter applies this slow-field and constitutive logic to neutral two-fluid materials; this page retains the generic hydrodynamic effective theory.

Solids and viscoelastic media

Broken translations introduce displacement or strain fields. Depending on defect motion and relaxation, a material can behave elastically at short times and hydrodynamically at long times. Ordinary fluid constitutive relations omit this memory.

Critical dynamics

Near a continuous transition, a long-lived order parameter and conserved densities can couple. Static universality alone does not fix the dynamic universality class. Whether the order parameter is conserved and which densities it couples to determine the appropriate Hohenberg–Halperin-type model.

Integrable systems

An integrable system can have an extensive set of conserved quasiparticle occupations. Generalized hydrodynamics evolves a distribution over species and rapidity rather than a short vector of charge densities. Integrability and Generalized Gibbs Ensembles Preview introduces these additional conserved distributions.

Localized systems

An ideal many-body-localized phase does not support ordinary thermal diffusion. Slow dephasing, rare-region effects, finite-size crossover, or external coupling can mimic unusual hydrodynamics over limited windows. Many-Body Localization Preview explains those limitations.

Causality, Stability, and Higher Orders

The diffusion equation has a Gaussian Green function with nonzero support at every distance for any $t>0$. This does not imply that a microscopic local quantum system transmits information instantaneously. Diffusion is an asymptotic equation valid for

$$q \ll \ell_{\mathrm{micro}}^{-1}, \qquad \omega \ll \tau_{\mathrm{micro}}^{-1}.$$

Its exponentially small far tail lies outside the regime in which the truncated equation resolves a microscopic front.

A simple model with a finite propagation speed introduces current relaxation:

$$\tau_J\partial_t\mathbf j + \mathbf j = -D\nabla n.$$

Together with continuity, this gives

$$\tau_J\partial_t^2n+\partial_tn=D\nabla^2n,\qquad \omega_\pm=\frac{-i\pm\sqrt{4D\tau_Jq^2-1}}{2\tau_J}.$$

For positive $D,\tau_J$, the two small-$q$ poles approach $-iDq^2$ and $-i/\tau_J$. The front speed is $\sqrt{D/\tau_J}$; relativistic causality additionally requires this speed not to exceed $c$. This equation is one possible transient model, not a unique microscopic completion. Below $\tau_J^{-1}$ the current can be eliminated; near that scale it must remain dynamical.

Relativistic first-order viscous equations can become acausal or unstable when extrapolated beyond their proper regime. Higher-order or transient formulations can repair some problems, but coefficient constraints and mode stability must be checked. “Second order” is not by itself a proof of causality or correctness.

Boundaries, Finite Size, and Initial Data

Hydrodynamics is a boundary-value and initial-value theory. A bulk equation does not determine:

• no-slip, partial-slip, or stress boundary conditions;
• absorbing, reflecting, or reservoir-coupled charge boundaries;
• thermal contact resistance;
• surface modes and edge currents;
• injection profiles;
• the projection of microscopic initial data onto slow fields.

For a finite closed region,

$$\frac{dQ_A}{dt} = -\int_{\partial V} dS_i\, j_A^i.$$

Global conservation follows only when the boundary flux vanishes. In a box of size $L$, the smallest nonzero wave number is of order $L^{-1}$, so a diffusive relaxation time scales as

$$\tau_L \sim \frac{L^2}{D}.$$

At times of order $L^2/D$, the infinite-medium decay law must be replaced by the finite-domain hydrodynamic modes and their boundary conditions. This does not invalidate hydrodynamics itself. Much later quantum recurrences or a lack of separation between local relaxation and the box scales can invalidate the irreversible description.

Quantum and Classical Fluctuation Regimes

At nonzero temperature and sufficiently low frequency,

$$\omega \ll T,$$

the KMS factor reduces to its classical form and a stochastic hydrodynamic action is often adequate. This does not make the microscopic system classical. It means the highly occupied low-frequency modes have an effectively classical fluctuation relation.

At $T=0$, near a quantum critical point, or at frequencies comparable to $T$, quantum fluctuation structure remains important. A classical white-noise action need not apply. Superfluids, topological phases, anomalies, and driven open systems can also require additional quantum or geometric data.

Coarse-grained entropy production is compatible with microscopic unitary evolution. Hydrodynamics discards detailed correlations in fast degrees of freedom and follows only a restricted set of observables. The underlying von Neumann entropy of a closed total state need not increase.

Relation to Renormalization and Matching

Hydrodynamics is a Wilsonian idea in real time: integrate out faster and shorter-distance structure while retaining all operators allowed by the symmetries and in-in constraints. Its cutoff satisfies

$$q \lesssim \Lambda_{\mathrm{hydro}} \ll \ell_{\mathrm{micro}}^{-1}.$$

Changing the cutoff shifts effective coefficients so predictions remain invariant to the working order. Nonlinear hydrodynamic loops can make this running explicit.

Near a critical point, the correlation length becomes an additional large scale and critical fluctuations must remain in the EFT. Critical Phenomena and RG Bridge develops the static scaling logic; dynamic critical field theory additionally tracks relaxation and conservation.

The distinction between matching and evolution is essential:

• microscopic theory or data determine the EFT coefficients at a scale;
• hydrodynamic equations evolve the retained fields below that scale;
• renormalization relates descriptions with different cutoffs;
• uncertainty from omitted operators sets the truncation error.

When Full Field Theory Is Needed

The many-body equations on this page are sufficient for identifying slow fields, leading pole structures, and the logic of matching. A full hydrodynamic QFT treatment becomes necessary for:

• nonlinear fluctuating hydrodynamics beyond tree level;
• systematic Schwinger–Keldysh operator bases;
• dynamical KMS and topological unitarity symmetries;
• relativistic fluids and causal transient theories;
• gauge, gravitational, or anomalous backgrounds;
• superfluids, solids, active matter, and reaction networks as complete EFTs;
• critical dynamics and dynamic renormalization groups;
• multi-point response, counting statistics, and noise interactions;
• controlled renormalization of long-time tails.

Continue with Quantum Field Theory suggests how to extend this calculation to nonlinear, relativistic, and fluctuating theories.

Method-Selection Guide

Static equilibrium

Use thermodynamics or Euclidean methods for equations of state and static susceptibilities. Hydrodynamic evolution is unnecessary unless a slow spacetime variation is part of the question.

Weak near-equilibrium response

Use Kubo formulas to match coefficients, then hydrodynamics to organize the joint low-$q$, low-$\omega$ response.

Dilute quasiparticle regime

Use Boltzmann transport when a controlled distribution function and collision integral exist. Its moments can match onto hydrodynamics after nonconserved distortions relax.

Strongly interacting long-distance regime

Hydrodynamics can remain valid without quasiparticles if local relaxation and a closed set of slow fields are established.

Finite coherent system

Direct unitary evolution, exact diagonalization, or tensor networks may be more informative before a scale-separated hydrodynamic window appears.

Open or driven system

Include explicit source, sink, noise, and steady-state structure. Ordinary equilibrium KMS and entropy arguments do not transfer automatically.

Integrable or nearly integrable system

Use generalized or quasihydrodynamic variables rather than forcing an extensive slow sector into one diffusion constant.

Exercises

Exercise 1: Select the slow fields

For each system, name the minimal hydrodynamic fields and one possible extra slow field:

1. a neutral translation-invariant normal fluid;
2. a dirty electronic conductor with strong momentum relaxation;
3. a neutral superfluid;
4. a system near a nonconserved Ising order-parameter transition;
5. a weakly nonintegrable one-dimensional gas.

Solution

1. Retain energy, mass or particle density, and momentum density. An additional internal charge is included if exactly conserved.
2. Retain charge and energy densities, with electrostatic screening or the electromagnetic field included consistently. Unscreened Coulomb interactions can turn a naive charge-diffusion mode into a plasma or charge-relaxation mode. Momentum can be integrated out only if its relaxation rate is fast compared with the working frequencies; otherwise it is quasihydrodynamic.
3. Retain energy, particle density, momentum, and the superfluid Goldstone phase.
4. Retain energy and any exact charges, plus the slowly relaxing Ising order parameter. Its nonconservation distinguishes its dynamic class from a conserved scalar.
5. Retain the usual exact densities and enough approximately conserved quasiparticle or charge distributions to describe the long prethermal window. Ordinary few-field hydrodynamics may emerge only after integrability-breaking relaxation.

Exercise 2: Diffusion from conservation

Combine

$$\partial_t n + \nabla\boldsymbol\cdot\mathbf j = 0$$

with $\mathbf j=-D\nabla n$. Find the pole of a plane-wave perturbation and explain why its rate vanishes as $q\to0$.

Solution

Substitution gives

$$\partial_t n = D\nabla^2n.$$

For

$$n \propto e^{i\mathbf q\cdot\mathbf x-i\omega t},$$

one obtains

$$-i\omega = -Dq^2,$$

so

$$\omega = -iDq^2.$$

The rate $Dq^2$ vanishes because a long-wavelength density excess cannot decay locally; the conserved quantity must flow over a distance of order $q^{-1}$.

Exercise 3: Ideal sound

Use the three ideal longitudinal equations

$$\begin{aligned} \partial_t\delta\rho &= -\rho_0\nabla\boldsymbol\cdot\mathbf v, \\ \rho_0\partial_t\mathbf v &= -\nabla\delta p, \\ \delta p &= c_s^2\delta\rho \end{aligned}$$

to derive the sound dispersion.

Solution

Differentiate the continuity equation with respect to time and insert the divergence of the momentum equation:

$$\begin{aligned} \partial_t^2\delta\rho &= -\rho_0 \nabla\boldsymbol\cdot \partial_t\mathbf v \\ &= \nabla^2\delta p \\ &= c_s^2\nabla^2\delta\rho. \end{aligned}$$

A plane wave therefore satisfies

$$\omega^2 = c_s^2q^2,$$

so

$$\omega_\pm = \pm c_sq.$$

Viscosity and thermal conduction add attenuation of order $-iq^2$ but do not change the leading ideal propagation speed.

Exercise 4: Weakly broken conservation

Let

$$\partial_t n + \nabla\boldsymbol\cdot\mathbf j = -\Gamma n, \qquad \mathbf j = -D\nabla n.$$

Here $n$ is a departure from the equilibrium density. Find the mode frequency and identify the regimes in which $n$ should remain an explicit field.

Solution

The equation becomes

$$\partial_t n = D\nabla^2n - \Gamma n.$$

For a plane wave,

$$\omega = -i \left( \Gamma+Dq^2 \right).$$

If $\Gamma$ is much smaller than microscopic relaxation rates and comparable to the frequencies of interest, $n$ is quasihydrodynamic and should remain explicit. When $|\omega|$ and $Dq^2$ are far below $\Gamma$, it can often be integrated out, though doing so changes the local operator expansion for the remaining fields.

Exercise 5: Coupled diffusion

Two densities obey

$$\partial_t \begin{pmatrix} n_1 \\ n_2 \end{pmatrix} = \begin{pmatrix} D_1 & \alpha \\ \beta & D_2 \end{pmatrix} \nabla^2 \begin{pmatrix} n_1 \\ n_2 \end{pmatrix}.$$

Find the two diffusion eigenvalues. What stability condition must they satisfy?

Solution

The matrix eigenvalues are

$$D_\pm = \frac{D_1+D_2}{2} \pm \frac12 \sqrt{ (D_1-D_2)^2 + 4\alpha\beta }.$$

The modes have

$$\omega_\pm = -iD_\pm q^2.$$

Stability requires nonnegative real parts and, for any zero-real-part eigenvalues, no nontrivial Jordan blocks producing secular growth. Strictly positive real parts give decay. For a real symmetric diffusion matrix, nonnegative eigenvalues reduce to

$$\begin{gathered} D_1\geq0, \qquad D_2\geq0, \\ \qquad D_1D_2-\alpha^2\geq0. \end{gathered}$$

In general the positivity test should be performed in the susceptibility-weighted thermodynamic variables, because $D=L\chi^{-1}$ need not be symmetric in the original density basis.

Exercise 6: Noise required by equilibrium

For a nonzero box-normalized Fourier mode $n_{\mathbf q}=V^{-1/2}\int_V d^dx\,e^{-i\mathbf q\cdot\mathbf x}n(\mathbf x)$, suppose

$$\dot n_{\mathbf q} = -Dq^2n_{\mathbf q} + \eta_{\mathbf q},$$

with white noise

$$\langle \eta_{\mathbf q}(t) \eta_{-\mathbf q}(t') \rangle = \mathcal A_q \delta(t-t').$$

Demand the equilibrium variance $\langle|n_{\mathbf q}|^2\rangle=T\chi$. Determine $\mathcal A_q$.

Solution

Let

$$C_q(t) = \langle|n_{\mathbf q}(t)|^2\rangle.$$

For an Ornstein–Uhlenbeck mode,

$$\frac{dC_q}{dt} = -2Dq^2C_q + \mathcal A_q.$$

Stationarity and $C_q=T\chi$ give

$$0 = -2Dq^2T\chi + \mathcal A_q.$$

Therefore

$$\mathcal A_q = 2T\chi Dq^2.$$

The factor $q^2$ shows that density noise enters through the divergence of a fluctuating current. At $q=0$, both terms vanish and this argument fixes no variance: a fixed-charge preparation has zero total-charge variance, whereas an initial mixture of charges keeps its original variance.

Exercise 7: Frame invariance

At first derivative order, redefine the velocity by $u^i\to u^i+\delta u^i_{(1)}$. Explain why individual constitutive coefficients can change while the pole locations of physical correlation functions cannot.

Solution

The redefinition changes how the exact current and stress are decomposed into “ideal” and “derivative” pieces. Terms proportional to $\delta u^i_{(1)}$ move between those pieces, so coefficients attached to a particular decomposition can change.

The full physical currents do not change. Correlation functions are obtained by differentiating with respect to external sources, and their poles are properties of the complete linearized equations. An invertible local field redefinition changes the coordinate description of those equations but not their physical eigenfrequencies to the retained order. A disagreement in a frame-dependent coefficient is meaningful only after both descriptions are translated to the same frame.

Exercise 8: Choose the sufficient theory

Choose among equilibrium thermodynamics, Kubo response, kinetic theory, ordinary hydrodynamics, quasihydrodynamics, generalized hydrodynamics, and full Schwinger–Keldysh EFT:

1. static compressibility;
2. dc conductivity from equilibrium current correlations;
3. dilute-gas viscosity with controlled binary collisions;
4. long-wavelength sound after local equilibration;
5. flow with a very slowly relaxing momentum;
6. an integrable gas with inhomogeneous quasiparticle occupations;
7. nonlinear noise vertices and long-time-tail loops.

Solution

1. Use equilibrium thermodynamics.
2. Use Kubo response with the correct limit order and contact terms.
3. Use kinetic theory, then match its moments to hydrodynamic coefficients if desired.
4. Use ordinary hydrodynamics with density, energy, and momentum fields.
5. Use quasihydrodynamics retaining momentum and its small relaxation rate.
6. Use generalized hydrodynamics.
7. Use a full fluctuating or Schwinger–Keldysh hydrodynamic EFT.

The options form a hierarchy, not a ranking. The sufficient theory is the one that retains the relevant slow information with controlled approximations.

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