Hydrodynamics Beyond the Gradient Expansion: Resurgence and Resummation

Abstract: Consistent formulations of relativistic viscous hydrodynamics involve short lived modes, leading to asymptotic rather than convergent gradient expansions. In this Letter we consider the Mueller-Israel-Stewart theory applied to a longitudinally expanding quark-gluon plasma system and identify hydrodynamics as a universal attractor without invoking the gradient expansion. We give strong evidence for the existence of this attractor and then show that it can be recovered from the divergent gradient expansion by Borel summation. This requires careful accounting for the short-lived modes which leads to an intricate mathematical structure known from the theory of resurgence.

• The paper establishes a universal hydrodynamic attractor using resurgence theory and Borel resummation techniques.
• It demonstrates that the traditional gradient expansion diverges at high orders, prompting the need for alternative methods.
• Utilizing MIS theory under Bjorken flow, the study refines the theoretical approach to quark-gluon plasma behavior.

Overview of "Hydrodynamics beyond the Gradient Expansion: Resurgence and Resummation"

The paper "Hydrodynamics beyond the Gradient Expansion: Resurgence and Resummation" by Michal P. Heller and Michał Spaliński explores the conceptual frameworks that underlie relativistic viscous hydrodynamics, with a specific focus on a universal attractor in the system dynamics, independent of the gradient expansion approach traditionally used. The authors investigate the Müller-Israel-Stewart (MIS) theory in the context of quark-gluon plasma systems, which are of significant interest due to their relevance in heavy ion collision experiments.

Key Concepts and Theoretical Insights

1. Relativistic Viscous Hydrodynamics and Quark-Gluon Plasma: The study emphasizes the importance of relativistic viscous hydrodynamics in describing quark-gluon plasma (QGP) in high-energy physics. Given the complexity of QGP and its behavior under experimental conditions at colliders like RHIC and LHC, the findings contribute substantially to theoretical physics, bridging gaps in understanding viscosity-related phenomena in QGP.
2. Gradient Expansion and Its Divergence: Hydrodynamics is usually depicted through a gradient expansion. However, this series expansion is asymptotic and diverges at high orders, especially in systems with significant fluctuations, such as strongly coupled ${\cal N} = 4$ SYM plasma. The non-convergent nature of this expansion underlines the need for alternative theoretical models.
3. Hydrodynamic Attractor and Resurgence Theory: The authors propose that hydrodynamics should not merely be seen as a gradient series but instead as an attractor within fluid dynamics. This attractor framework accounts for the exponential decay of nonhydrodynamic modes, leading to an evolved understanding of hydrodynamics as a universal attractor. The attractor's existence is validated using the Borel summation technique and resurgence theory principles, describing how divergent series can be mathematically resumed.
4. Müller-Israel-Stewart Theory and Its Application: The paper uses the MIS theory, which introduces dynamical fields for shear stress tensors to maintain causality, as its central model. MIS is explored under Bjorken flow conditions to manage complexities inherent in the governing equations, making the attractor identification feasible.

Numerical Results and Analytical Methods

• Hydrodynamic Attractor Verification: Through numerical simulations and analysis, the hydrodynamic attractor emerges as robust and detachable from initial conditions. The attractor continues to manifest even when traditional hydrodynamic truncations fail to be adequate.
• Gradient Expansion Terms: The paper provides a deep dive into the high-order behavior of the hydrodynamic gradient expansion, confirming its asymptotic divergence through calculated coefficients indicative of factorial growth.
• Borel Resummation and Padé Approximants: A considerable portion of the work deploys Borel transform and Padé approximants to continue the analytical structure of the divergent series. This resummation aligns the function's behavior with the hydrodynamic attractor.

Implications and Future Directions

This work suggests a compelling shift in the understanding of relativistic hydrodynamics, with implications significant for phenomenological models of QGP. By identifying a unique attractor solution as fundamental to understanding fluid dynamics beyond the naive gradient expansion, the study opens pathways for reinterpretations of heavy ion collisions and effective field theories. The approach also highlights the broader utility of resurgence theory in complex quantum systems, thus pointing to potential applications in AdS/CFT correspondences and beyond.

Future research could aim to apply these findings to other theoretical and experimental setups, possibly extending the methodology to explore other nontrivial systems in high-energy physics. The synergy between theoretical advancements in hydrodynamics and experimental practices promises a refined predictive toolset for upcoming challenges in the field.