Nonlinear Fluctuating Hydrodynamics

• Nonlinear fluctuating hydrodynamics is a mesoscopic theory that expands Euler currents to quadratic order to describe anomalous transport via coupled stochastic equations.
• The theory employs a normal mode transformation to derive coupled Burgers equations, predicting diverse scaling regimes such as KPZ, Lévy, and diffusive behavior.
• Applications to anharmonic chains, spin models, and quantum systems illustrate the framework’s versatility in capturing complex transport phenomena with mode-coupling approximations.

Nonlinear fluctuating hydrodynamics (NLFH) is a mesoscopic theory for systems with local conservation laws in which the Euler currents are kept to second order in the deviations from equilibrium and dissipation plus noise are added. In one spatial dimension, where linear fluctuating hydrodynamics is insufficient because transport is often anomalous, this nonlinear extension leads to coupled stochastic conservation laws whose long-time correlations can display Kardar-Parisi-Zhang (KPZ), Lévy, diffusive, or related scaling, depending on the conservation laws and the nonlinear mode couplings (Spohn, 2013, Spohn, 2015, Schütz, 2017).

1. Conceptual and mathematical framework

The canonical NLFH construction starts from locally conserved fields $u_\alpha(x,t)$ obeying conservation laws of the form

$$\partial_t u_\alpha(x,t) + \partial_x j_\alpha(u(x,t)) = 0.$$

Expanding the currents around equilibrium to quadratic order, and then adding dissipation and stochastic forcing, yields a nonlinear stochastic field theory. For anharmonic chains this is written as

$$\partial_t u(x,t) + \partial_x \left[ Au(x,t) + \frac{1}{2} (u(x,t), H u(x,t)) \right] = \partial_x (D \partial_x u(x,t)) + \partial_x(B \xi(x,t)),$$

with $A$ the linearized Euler matrix, $H$ the Hessians of the currents, $D$ a positive-definite diffusion matrix, and $\xi$ vector-valued space-time white noise (Spohn, 2013).

A central step is the transformation to normal modes. If $R$ diagonalizes $A$, then $\phi = Ru$ separates the linear propagation into modes with velocities $$\partial_t u_\alpha(x,t) + \partial_x j_\alpha(u(x,t)) = 0.$$0. In that basis, the equations take the coupled stochastic Burgers form

$$\partial_t u_\alpha(x,t) + \partial_x j_\alpha(u(x,t)) = 0.$$1

where the tensors $$\partial_t u_\alpha(x,t) + \partial_x j_\alpha(u(x,t)) = 0.$$2 encode quadratic mode coupling (Spohn, 2015). For a single conserved mode this reduces to the noisy Burgers equation, equivalently the spatial derivative of the KPZ equation; for several modes no exact solutions are known in general, and one-loop mode-coupling theory supplies the standard approximation scheme (Mendl et al., 2013).

The mode-coupling approximation is based on the empirical decoupling of separated peaks at long times. In Fourier space it yields integro-differential equations with quadratic memory kernels. For the diagonal approximation,

$$\partial_t u_\alpha(x,t) + \partial_x j_\alpha(u(x,t)) = 0.$$3

with

$$\partial_t u_\alpha(x,t) + \partial_x j_\alpha(u(x,t)) = 0.$$4

This structure is the basis for the large-scale predictions of NLFH in one-dimensional systems with several conserved fields (Mendl et al., 2013).

2. Anharmonic chains and the three-mode scenario

The standard laboratory for NLFH is the one-dimensional anharmonic chain, whose locally conserved fields are stretch, momentum, and energy. Spohn’s formulation for anharmonic chains makes the model-dependent parameters explicit in terms of the interaction potential, pressure, and temperature, and develops the resulting nonlinear stochastic field theory in the one-loop approximation (Spohn, 2013).

In normal-mode coordinates, the generic equilibrium structure consists of two propagating sound modes and one stationary heat mode. The asymptotic prediction is that the sound peaks broaden with KPZ scaling,

$$\partial_t u_\alpha(x,t) + \partial_x j_\alpha(u(x,t)) = 0.$$5

while the heat peak is generically broader and follows a Lévy $$\partial_t u_\alpha(x,t) + \partial_x j_\alpha(u(x,t)) = 0.$$6 law,

$$\partial_t u_\alpha(x,t) + \partial_x j_\alpha(u(x,t)) = 0.$$7

These statements appear both in the original development of NLFH for anharmonic chains and in the later review-style exposition focused on equilibrium time correlations (Mendl et al., 2013, Spohn, 2015).

The physical significance of the three-mode picture is that anomalous transport is not inserted phenomenologically at the level of scaling exponents; it is generated by the quadratic part of the Euler currents. Linear fluctuating hydrodynamics would predict Gaussian broadening, whereas the nonlinear theory predicts that the sound peaks fall in the KPZ universality class whenever the self-coupling $$\partial_t u_\alpha(x,t) + \partial_x j_\alpha(u(x,t)) = 0.$$8, and that the heat mode spreads superdiffusively through mode coupling to the sound peaks (Spohn, 2015).

This framework also yields predictions for total current correlations and for dynamical phase diagrams. Exceptional symmetry situations can alter the generic three-mode scenario. The data summarized for anharmonic chains explicitly note that special symmetry situations can lead to diffusive sound and different heat scaling, rather than the generic KPZ-sound/Lévy-heat combination (Spohn, 2015).

3. Universality classes and exact tests

NLFH does not predict a single universal exponent. In one dimension, the possible dynamical universality classes within its range of applicability can form an infinite discrete family whose exponents are Kepler ratios of neighboring Fibonacci numbers,

$$\partial_t u_\alpha(x,t) + \partial_x j_\alpha(u(x,t)) = 0.$$9

including diffusion with $$\partial_t u(x,t) + \partial_x \left[ Au(x,t) + \frac{1}{2} (u(x,t), H u(x,t)) \right] = \partial_x (D \partial_x u(x,t)) + \partial_x(B \xi(x,t)),$$0, KPZ with $$\partial_t u(x,t) + \partial_x \left[ Au(x,t) + \frac{1}{2} (u(x,t), H u(x,t)) \right] = \partial_x (D \partial_x u(x,t)) + \partial_x(B \xi(x,t)),$$1, and the limiting golden mean class (Schütz, 2017). The decisive input is the vanishing pattern of the diagonal mode-coupling elements $$\partial_t u(x,t) + \partial_x \left[ Au(x,t) + \frac{1}{2} (u(x,t), H u(x,t)) \right] = \partial_x (D \partial_x u(x,t)) + \partial_x(B \xi(x,t)),$$2, which is itself determined by the stationary current-density relations.

The case of two conserved fields was worked out in detail for the BS model. There the two normal modes are a traveling sound mode and a standing heat mode, and NLFH predicts a KPZ scaling function for the moving peak and a maximally asymmetric Lévy distribution with parameter $$\partial_t u(x,t) + \partial_x \left[ Au(x,t) + \frac{1}{2} (u(x,t), H u(x,t)) \right] = \partial_x (D \partial_x u(x,t)) + \partial_x(B \xi(x,t)),$$3 for the stationary peak (Spohn et al., 2014). The same work provides a complete classification of universality classes for two coupled stochastic Burgers equations with arbitrary coupling coefficients, including modified KPZ and diffusive cases.

A notable analytic confirmation of multicomponent NLFH was obtained for the two-species Arndt-Heinzl-Rittenberg exclusion process. For step-type initial data, the exact long-time joint current distribution becomes a product of a Gaussian and a GUE Tracy-Widom distribution, providing the first analytic confirmation for a multi-component system of a prediction from nonlinear fluctuating hydrodynamics for one-dimensional systems (Chen et al., 2018).

A different exacting test arises in the discrete Gross-Pitaevskii equation. There the low-temperature dynamics can be rewritten in hydrodynamic variables and then mapped to a nonlinear fluctuating hydrodynamics theory whose chiral normal modes reduce to noisy Burgers equations. The resulting dynamical structure factor exhibits the KPZ line width $$\partial_t u(x,t) + \partial_x \left[ Au(x,t) + \frac{1}{2} (u(x,t), H u(x,t)) \right] = \partial_x (D \partial_x u(x,t)) + \partial_x(B \xi(x,t)),$$4, and the mapping is benchmarked against exact Hamiltonian numerics (Kulkarni et al., 2015). Taken together, these results show that NLFH organizes both integrable-style exact asymptotics in special stochastic models and effective stochastic descriptions of chaotic Hamiltonian systems.

4. Approximate conservation, crossover, and departures from the canonical picture

A recurring theme in NLFH is that the effective number of conserved fields can change with temperature, time scale, or symmetry. Chains of nonlinearly coupled rotators provide a paradigmatic example. Their Hamiltonian is

$$\partial_t u(x,t) + \partial_x \left[ Au(x,t) + \frac{1}{2} (u(x,t), H u(x,t)) \right] = \partial_x (D \partial_x u(x,t)) + \partial_x(B \xi(x,t)),$$5

and they conserve angular momentum and energy, but not true stretch, because the angular differences are defined modulo $$\partial_t u(x,t) + \partial_x \left[ Au(x,t) + \frac{1}{2} (u(x,t), H u(x,t)) \right] = \partial_x (D \partial_x u(x,t)) + \partial_x(B \xi(x,t)),$$6. As a result, the equilibrium Euler currents vanish,

$$\partial_t u(x,t) + \partial_x \left[ Au(x,t) + \frac{1}{2} (u(x,t), H u(x,t)) \right] = \partial_x (D \partial_x u(x,t)) + \partial_x(B \xi(x,t)),$$7

and NLFH predicts Gaussian diffusive correlations with two uncoupled diffusive modes at intermediate temperatures (Spohn, 2014). At very low temperature, however, the system rarely winds by $$\partial_t u(x,t) + \partial_x \left[ Au(x,t) + \frac{1}{2} (u(x,t), H u(x,t)) \right] = \partial_x (D \partial_x u(x,t)) + \partial_x(B \xi(x,t)),$$8, so it temporarily behaves like an FPU chain with an effective third conserved field, showing two KPZ sound peaks and a strongly suppressed stationary heat peak before crossing over to the true diffusive regime (Spohn, 2014).

The classical XXZ spin chain shows an analogous low-temperature mechanism in the easy-plane regime. Energy and $$\partial_t u(x,t) + \partial_x \left[ Au(x,t) + \frac{1}{2} (u(x,t), H u(x,t)) \right] = \partial_x (D \partial_x u(x,t)) + \partial_x(B \xi(x,t)),$$9-magnetization are exactly conserved, while the difference between neighboring spin angles in the $A$0 plane becomes an almost conserved field because phase slips are rare. NLFH then predicts a heat peak and propagating sound peaks with anomalous broadening; the molecular-dynamics tests reported for the classical XXZ chain find two sound peaks with KPZ scaling in a suitable intermediate temperature regime, while at high temperature both easy-plane and easy-axis cases show diffusive spin and energy peaks and no sound modes (Das et al., 2018).

The validity of the standard NFHD scenario has also been challenged from the molecular-dynamics side. An effective mode-detection scheme that explicitly incorporates microscopic pressure fluctuations led to the conclusion that NFHD predictions for anharmonic chains are valid only when the pressure is zero and the pressure fluctuations are weak, and that for nonvanishing pressure at least three universality classes of transport should be distinguished (Xiong, 2019). This does not negate the standard mode-coupling theory, but it narrows the regimes in which its textbook decoupling picture is claimed to hold.

A related crossover problem was analyzed in a hard-particle gas with stochastic three-particle collisions. There the diffusion and noise terms of fluctuating hydrodynamics are derived microscopically rather than added phenomenologically, and the stationary energy current takes the form

$A$1

combining the anomalous $A$2 contribution with a diffusive $A$3 correction and an explicit microscopic crossover length $A$4 (Miron et al., 2018).

5. Microscopic derivations and transport coefficients

Although NLFH was first formulated as an effective stochastic theory, several works derive its ingredients from microscopic dynamics. A microscopic theory for nonlinear lattices combines coarse-graining with projection-operator techniques and emphasizes the critical role of equivalence of ensembles. In that construction, the chain is divided into large blocks, the dynamics is projected onto coarse-grained conserved fields, and a Green-Kubo-like formula for the bare transport coefficients is obtained in a numerically computable form. Numerical simulations then show that these bare transport coefficients exist for a sufficiently large but finite coarse-graining length in the infinite lattice, supporting the claim that they uniquely exist for each physical system (Saito et al., 2020).

For crystalline solids, the slow variables must include the displacement field because translation symmetry is spontaneously broken. A microscopic derivation from a many-particle Hamiltonian proposes a microscopic expression for the displacement field,

$A$5

and derives nonlinear mode-coupling terms in the reversible currents that agree with phenomenological nonlinear fluctuating hydrodynamics. The derivation relies on projection onto coarse-grained fields, long-wavelength expansion, and the stationarity condition of the Fokker-Planck equation (Hiura, 2023).

For driven diffusive systems with bulk dissipation, fluctuating hydrodynamics takes a different balance-law form,

$A$6

with

$A$7

The quasi-elastic scaling of the microscopic dynamics makes the dissipation noise subdominant, so stochasticity enters through the current noise, and the microscopic complexity is condensed into the diffusivity, mobility, and dissipation coefficient (Prados et al., 2012). This extension shows that NLFH-type reasoning is not restricted to conservative Hamiltonian chains.

6. Extensions across dimensions, ordered media, and critical or strongly coupled systems

NLFH has been extended well beyond the original three-field one-dimensional chain. For a chain vibrating in three spatial dimensions with first-neighbor anharmonic interactions and external tension, the conserved fields comprise three elongations, three momenta, and energy, yielding seven normal modes: one heat mode, two longitudinal sound modes, and four transverse sound modes (Barreto et al., 2019). In the reported NLFHT analysis and molecular-dynamics comparison, the transverse sound modes are diffusive with Gaussian scaling, the longitudinal sound modes are superdiffusive with KPZ scaling, and the heat mode is superdiffusive with an exponent between $A$8 and $A$9, depending on coupling to the sound sectors (Barreto et al., 2019).

Critical conservative dynamics provides another extension. For the one-dimensional fluctuating Ising-Kac-Kawasaki equation in a near-critical scaling regime, a sequence of rescaled conservative SPDEs converges to the stochastic Cahn-Hilliard equation,

$H$0

thereby establishing a rigorous nonlinear fluctuating-hydrodynamic limit in a regime where Gaussian fluctuation theory is replaced by nonlinear fluctuation behavior (Wu, 19 Nov 2025). The same work proves a multi-scale dynamical large deviation principle and $H$1-convergence of the prelimit rate functions to the Cahn-Hilliard rate function.

Strongly coupled quantum many-body systems can also be reorganized into fluctuating hydrodynamics. For the SYK lattice, the low-temperature dynamics is dominated by pseudo-Goldstone reparametrization modes. In the long-wavelength limit, the nonlinear action of these modes can be reorganized as the effective field theory for fluctuating hydrodynamics, and the hydrodynamic effective action is computed to high orders in the derivative expansion with all corresponding transport coefficients determined microscopically (Bucca et al., 20 Apr 2026). A plausible implication is that NLFH can function not only as an emergent phenomenology but also as a systematically derived effective field theory in certain strongly coupled settings.

7. Relativistic formulations and current debates

Relativistic fluctuating hydrodynamics requires additional structural input because causality forces dissipative currents to become dynamical variables. In one formulation, constitutive relations are written in integral form with memory functions, so the noises are colored by the fluctuation-dissipation relation. Under the assumption of Gaussian noise, however, the integral equations can be rewritten as differential stochastic equations with white noise, which is advantageous for numerical implementation (Murase et al., 2013).

A later effective-theory construction develops nonlinear stochastic relativistic hydrodynamics within divergence-type hydrodynamics. The equations are written as

$H$2

with entropy current

$H$3

By imposing a symmetry of the effective action derived from the Crooks fluctuation theorem,

$H$4

the resulting equations are flux-conservative and symmetric hyperbolic when the dynamics is causal, so the theory is mathematically well posed for suitable initial data (Mullins et al., 1 Oct 2025).

At the level of one-dimensional NLFH itself, there is an active disagreement over the asymptotic heat mode. The standard mode-coupling scenario for anharmonic chains predicts KPZ sound peaks together with a Lévy-broadened heat peak (Spohn, 2013). By contrast, a symmetry-based formulation for one-dimensional many-particle systems with homogeneous nearest-neighbor interactions derives the hydrodynamic equations from symmetry and conservation principles and, using the dynamic renormalization group, concludes that both sound and heat modes share the dynamical exponent $H$5 and are close to the Prähofer-Spohn KPZ scaling function (Minami et al., 16 Nov 2025). This suggests a direct tension between symmetry-based RG and the conventional mode-coupling classification, rather than a settled consensus.

A second debate concerns scope rather than exponent values. The pressure-fluctuation study of anharmonic chains argues that the standard NFHD predictions are reliable only at zero pressure with weak pressure fluctuations, while other pressure regimes generate additional relaxation patterns and at least three universality classes (Xiong, 2019). Accordingly, NLFH is best understood not as a single closed doctrine, but as a family of effective descriptions whose conservation laws, mode couplings, symmetry constraints, and admissible asymptotic regimes remain under active refinement.