Relativistic Hydrodynamics: A Singulant Perspective

Abstract

There is growing evidence that the hydrodynamic gradient expansion is factorially divergent. We advocate for using Dingle's singulants as a way to gain analytic control over its large-order behaviour for nonlinear flows. Within our approach, singulants can be viewed as new emergent degrees of freedom which reorganise the large-order gradient expansion. We work out the physics of singulants for longitudinal flows, where they obey simple evolution equations which we compute in M\"uller-Israel-Stewart-like models, holography and kinetic theory. These equations determine the dynamics of the large-order behaviour of the hydrodynamic expansion, which we confirm with explicit numerical calculations. One of our key findings is a duality between singulant dynamics and a certain linear response theory problem. Finally, we discuss the role of singulants in optimal truncation of the hydrodynamic gradient expansion. A by-product of our analysis is a new M\"uller-Israel-Stewart-like model, where the qualitative behaviour of singulants shares more similarities with holography than models considered hitherto.