A new effective theory for stochastic relativistic hydrodynamics

TL;DR

This work addresses the need for a mathematically well-posed framework for stochastic relativistic hydrodynamics, relevant to the quark-gluon plasma near critical points and in small systems. It develops an effective action combining divergence-type hydrodynamics with the Crooks fluctuation theorem, parameterized by a generating current $X^{\mu}$ and a dissipative potential $\zeta_0^{ab}(\phi)$, ensuring flux-conservative, causal, and thermodynamically consistent evolution. The leading Lagrangian enforces a covariant Lyapunov functional and nonlinear fluctuation-dissipation relations, with a concrete relativistic diffusion example that couples current conservation to noisy relaxation of dissipative modes. This framework provides a practical, stable, and foliation-independent approach for simulating fluctuating relativistic fluids, offering a solid basis for incorporating fluctuations into gradient-expansion theories and broader applications in high-energy nuclear physics.

Abstract

Thermal fluctuations are a fundamental feature of dissipative systems that are essential for understanding physics near the expected critical point of QCD and in small systems. When such fluctuations are modeled naively in relativistic systems, strange features can appear such as negative self-correlation functions. We construct an effective theory for nonlinear stochastic relativistic hydrodynamics that ensure a well-posed mathematical formulation. Using Crooks fluctuation theorem, we derive a symmetry of the effective action that incorporates fluctuations through a suitable free energy functional. For divergence type theories, the action can then be fully specified using a single vector generating current. The equations of motion obtained using this procedure are guaranteed to be flux conservative and symmetric hyperbolic when the dynamics is causal. This ensures that these equations are well-posed (for suitable initial data) and are in a form that can easily be simulated, including with Metropolis techniques.