Nonlinear Stochastic Relativistic Hydrodynamics

• Nonlinear stochastic relativistic hydrodynamics is a framework that integrates nonlinear closure and stochastic fluctuations to model the evolution of relativistic fluids, particularly in heavy-ion collisions and near-critical systems.
• It employs divergence type theories and covariant effective actions to ensure local conservation laws, thermodynamic consistency, and mathematical well-posedness.
• Advanced simulation techniques, such as Metropolis algorithms, enable the accurate capture of non-Gaussian fluctuations and critical dynamics in far-from-equilibrium regimes.

Nonlinear stochastic relativistic hydrodynamics is a theoretical and computational framework that describes the macroscopic evolution of relativistic fluids, incorporating both nonlinearities from strong gradients and non-equilibrium effects, as well as intrinsic stochastic fluctuations arising from finite temperature or proximity to critical points. This construct is essential in high-energy physics, astrophysics, and cosmology, particularly for the description of the quark-gluon plasma in heavy-ion collisions and in systems near the QCD critical point. The modern formulation unifies structure from relativistic kinetic theory, principles of nonequilibrium statistical mechanics, covariant effective field theory, fluctuation-dissipation theorems, and new rigorous approaches to mathematical well-posedness.

1. Foundations and Motivation

Relativistic hydrodynamics extends the conservation laws for energy-momentum and conserved charges to systems exhibiting Lorentz invariance. In the deterministic setting, ideal and dissipative (Navier-Stokes or Israel-Stewart-type) relativistic hydrodynamics suffice for linear response and weakly out-of-equilibrium scenarios. However, realistic applications demand the inclusion of nonlinear corrections and the effects of thermal fluctuations. This is crucial near the QCD critical point, in small systems, and in regimes with strong gradients or far-from-equilibrium distributions, where standard gradient expansions break down (Peralta-Ramos et al., 2012, Murase et al., 2013, An et al., 2022, Mullins et al., 8 Jan 2025).

Stochasticity emerges from finite coarse-grained averaging over microscopic degrees of freedom (“hydrodynamic noise”), while nonlinearities arise from (i) the inherent nonlinear structure of conservation laws; (ii) nonlinear closure schemes connecting macroscopic and kinetic theory variables; and (iii) strong coupling to higher-order or nonhydrodynamic modes.

2. Nonlinear Closure and Divergence Type Theories

Beyond linearized or perturbative approaches, a key advance is the development of nonlinear closure schemes and divergence type theories (DTTs) that extend hydrodynamics into genuinely nonequilibrium regimes while preserving thermodynamic consistency.

• The nonequilibrium effective theory (NET) formalism introduces a dynamical nonequilibrium tensor γ^{μν}, which parameterizes deviations from equilibrium in the one-particle distribution function f via a closure obtained by extremizing entropy production (Entropy Production Principle, EPP) (Peralta-Ramos et al., 2012). The closure is genuinely nonlinear:

$$f = f_0 \left[ 1 + \frac{\tau_r}{TF} p^i p^j \gamma_{ij} + \frac{\tau_r^2}{2T^2F^2} p^i p^j p^l p^m \gamma_{ij} \gamma_{lm} \right]$$

The resulting evolution equations, derived by the method of moments, remain robust for large gradients and strong off-equilibrium, outperforming second order fluid dynamics in tracking Boltzmann equation solutions.

• Divergence type theories formulate all evolution equations as local conservation laws

$$\nabla_\mu \frac{\partial X^\mu}{\partial \phi^A} = I_A(\phi)$$

where $X^\mu$ is a vector-valued generating current determined by the equation of state and its dissipative extensions, and $I_A(\phi)$ encodes source terms (often chosen as linear in dissipative variables with a positive semi-definite coefficient matrix) (Mirón-Granese et al., 2020, Mullins et al., 1 Oct 2025). This formulation ensures exact flux conservation and nonperturbative compliance with the second law.

• Nonlinearities are essential not only in the closure relations but throughout: in the construction of the generating current, in the evolution of slow modes (e.g., specific entropy near criticality), and in the feedback between hydrodynamic and nonhydrodynamic sectors.

3. Incorporating Stochastic Fluctuations: Fluctuation-Dissipation and Covariant Effective Actions

Stochastic hydrodynamics introduces noise sources into dissipative and conserved current equations, governed by fluctuation-dissipation relations:

$$\langle \xi_A(x)\, \xi_B(x') \rangle = 2 \sigma_{AB} \delta^{(4)}(x - x')$$

where $\sigma_{AB}$ is the Onsager matrix capturing dissipation (Murase et al., 2013, Mullins et al., 2023, Mullins et al., 8 Jan 2025).

A modern, systematized way to achieve this is via a covariant effective action, constructed using path integrals (Martin-Siggia-Rose or Schwinger-Keldysh techniques) and auxiliary fields. The effective Lagrangian incorporates the dissipative and noise terms, with crucial constraints from the Crooks fluctuation theorem (Mullins et al., 8 Jan 2025, Mullins et al., 1 Oct 2025):

$$\mathcal{L}[\Theta \phi, \Theta \bar{\phi} - i \Theta \phi] = \mathcal{L}[\phi, \bar{\phi}] - i \sigma$$

where $\Theta$ is a time-reversal operation and $\sigma$ the local entropy production density. This $\mathbb{Z}_2$ symmetry ensures nonlinear fluctuation-dissipation relations for the full hierarchy of $n$-point functions and suppresses unphysical features (such as negative self-correlation functions).

The action is organized around a free energy (or entropy) functional, often manifest in the form of a Lyapunov function, which structures both dynamics and stochasticity:

$$\mathcal{L} = - \bar{\phi}^A \nabla_\mu \left( \frac{\partial X^\mu}{\partial \phi^A} \right) + i \zeta_0^{ab}(\phi) \bar{\phi}_a (\bar{\phi}_b + i \phi_b)$$

where the second term enforces the correct noise correlations.

4. Mathematical Well-Posedness, Causality, and Hyperbolicity

A rigorous theory of stochastic relativistic hydrodynamics requires mathematical well-posedness: symmetric hyperbolic evolution equations, causality (signals propagate inside the light cone), and local stability.

This is guaranteed by analyzing the characteristic matrix $$M^\mu_{ab} = \partial^2 X^\mu / (\partial \Phi^a \partial \Phi^b)$$ and ensuring that,

• $M^0$ is positive definite (enforces symmetric hyperbolicity and well-posedness),
• the principal symbol dictates characteristic speeds less than or equal to the speed of light,
• the Lyapunov functional is strictly convex and entropy production non-negative (Bemfica et al., 2019, Mullins et al., 8 Jan 2025, Mullins et al., 1 Oct 2025).

This is independent of spacetime foliation and holds in full generality, including coupling to gravity.

5. Evolution of Fluctuations: Moments, Non-Gaussianity, and Critical Dynamics

The framework treats both Gaussian and higher-order (non-Gaussian) fluctuations through evolution equations for cumulants or connected correlation functions:

$$u \cdot \partial G^{(c)}_{i_1 \dots i_n}(x_1, \dotsc, x_n) = \text{Drift terms} + \text{Noise terms} + \text{Mode-coupling terms}$$

For critical dynamics (e.g., near the QCD critical point), hierarchies of relaxation times emerge, with slowest-relaxing diffusive modes (such as specific entropy) controlling the non-Gaussian fluctuation dynamics (An et al., 2022, An et al., 28 Feb 2024). Covariant evolution equations for these multipoint functions have been constructed using a confluent formalism and Wigner transforms, allowing careful treatment of flow gradients, pressure coupling, and local Lorentz covariance.

Rotating wave approximation (RWA) techniques can be applied to simplify the evolution of high-dimensional multipoint correlators, isolating slow subspaces where nonlinear stochastic effects are most pronounced.

6. Numerical Simulation and Metropolis Algorithms

Well-posedness and the divergence type structure permit the application of robust simulation methods. Metropolis algorithms have been successfully implemented:

• Evolution is operator-split: an ideal hydrodynamics step followed by random momentum or charge transfers between neighboring cells,
• Each proposal is accepted/rejected according to the change in entropy (ΔS), ensuring detailed balance and the correct equilibrium fluctuation-dissipation relation (Basar et al., 7 Mar 2024, Bhambure et al., 13 Dec 2024, Mullins et al., 1 Oct 2025).
• The method directly incorporates detailed balance and is compatible with large time steps, unlike traditional Langevin approaches, and avoids the need for auxiliary relaxation variables.

Table: Key Simulation Elements

Element	Description	Reference
Frame	Density frame (local J^0 sets μ, T, u^μ)	(Bhambure et al., 13 Dec 2024Basar et al., 7 Mar 2024)
Update acceptance	Metropolis (ΔS as weight, exp(ΔS) for ΔS<0)	(Bhambure et al., 13 Dec 2024Basar et al., 7 Mar 2024)
Noise structure	Derived from effective action, Lyapunov stable	(Mullins et al., 1 Oct 2025Mullins et al., 8 Jan 2025)
Flux conservation	Guaranteed by divergence type theory	(Mullins et al., 1 Oct 2025Mirón-Granese et al., 2020)

This approach is conceptually and numerically robust for heavy-ion collision modeling, critical phenomena simulations, and studies of qausi-relativistic stochastic fluid dynamics.

7. Connections, Limitations, and Ongoing Developments

This field bridges relativistic kinetic theory, classical and quantum field theory approaches (Martin-Siggia-Rose, Schwinger-Keldysh), and rigorous mathematical analysis of PDEs. Current developments focus on:

• Extensions to full non-Gaussian fluctuations, higher-order cumulant evolution, and critical behavior (An et al., 2022An et al., 28 Feb 2024),
• Refinements of effective field theory with built-in fluctuation theorems and symmetry constraints (Mullins et al., 8 Jan 2025Mullins et al., 1 Oct 2025),
• The interplay between transport coefficient bounds, nonanalytic mode behavior, and macroscopic causality constraints (Heller et al., 2023),
• New simulation techniques that maintain well-posedness and detailed balance even in strongly nonlinear, out-of-equilibrium regimes (Bhambure et al., 13 Dec 2024).

Limitations remain regarding manifestly Lorentz-invariant discretization schemes, the treatment of sub-dissipative (ballistic) scales, and the coupling to full quantum many-body dynamics. However, the new systematic effective action and divergence type structures, underpinned by fluctuation theorems, provide a solid foundation for further advances and applications.

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In summary, nonlinear stochastic relativistic hydrodynamics is a mathematically rigorous and physically consistent framework that unifies nonlinear closure theory, divergence type dynamics, covariant fluctuation-dissipation, and modern effective action perspectives, with robust implications for the analysis and simulation of relativistic fluids far from equilibrium or near criticality (Peralta-Ramos et al., 2012, Murase et al., 2013, An et al., 2022, An et al., 28 Feb 2024, Bhambure et al., 13 Dec 2024, Mullins et al., 8 Jan 2025, Mullins et al., 1 Oct 2025).